| L(s) = 1 | + 18·5-s − 2·9-s + 6·13-s + 141·25-s − 18·29-s + 18·41-s − 36·45-s + 9·49-s − 18·61-s + 108·65-s − 3·81-s − 48·89-s − 6·97-s − 54·101-s + 30·113-s − 12·117-s − 27·121-s + 594·125-s + 127-s + 131-s + 137-s + 139-s − 324·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 8.04·5-s − 2/3·9-s + 1.66·13-s + 28.1·25-s − 3.34·29-s + 2.81·41-s − 5.36·45-s + 9/7·49-s − 2.30·61-s + 13.3·65-s − 1/3·81-s − 5.08·89-s − 0.609·97-s − 5.37·101-s + 2.82·113-s − 1.10·117-s − 2.45·121-s + 53.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 26.9·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(28.88124047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(28.88124047\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 2 T^{2} + 7 T^{4} + 4 p T^{6} + 7 p^{2} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12} \) |
| good | 5 | \( ( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{6} \) |
| 7 | \( 1 - 9 T^{2} - 9 T^{4} + 832 T^{6} - 3699 T^{8} - 16119 T^{10} + 317958 T^{12} - 16119 p^{2} T^{14} - 3699 p^{4} T^{16} + 832 p^{6} T^{18} - 9 p^{8} T^{20} - 9 p^{10} T^{22} + p^{12} T^{24} \) |
| 11 | \( 1 + 27 T^{2} + 459 T^{4} + 3316 T^{6} - 13419 T^{8} - 832383 T^{10} - 12002898 T^{12} - 832383 p^{2} T^{14} - 13419 p^{4} T^{16} + 3316 p^{6} T^{18} + 459 p^{8} T^{20} + 27 p^{10} T^{22} + p^{12} T^{24} \) |
| 13 | \( ( 1 - 3 T + 15 T^{2} - 36 T^{3} + 93 T^{4} - 753 T^{5} + 646 T^{6} - 753 p T^{7} + 93 p^{2} T^{8} - 36 p^{3} T^{9} + 15 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 + 27 T^{2} + 36 T^{3} + 27 p T^{4} + p^{3} T^{6} )^{4} \) |
| 19 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{6} \) |
| 23 | \( 1 - 21 T^{2} - 1077 T^{4} + 12164 T^{6} + 1038861 T^{8} - 242505 p T^{10} - 561401058 T^{12} - 242505 p^{3} T^{14} + 1038861 p^{4} T^{16} + 12164 p^{6} T^{18} - 1077 p^{8} T^{20} - 21 p^{10} T^{22} + p^{12} T^{24} \) |
| 29 | \( ( 1 + 9 T + 51 T^{2} + 216 T^{3} - 147 T^{4} - 7785 T^{5} - 54754 T^{6} - 7785 p T^{7} - 147 p^{2} T^{8} + 216 p^{3} T^{9} + 51 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 31 | \( 1 - 69 T^{2} + 2235 T^{4} - 26636 T^{6} - 1141011 T^{8} + 84778065 T^{10} - 3302984514 T^{12} + 84778065 p^{2} T^{14} - 1141011 p^{4} T^{16} - 26636 p^{6} T^{18} + 2235 p^{8} T^{20} - 69 p^{10} T^{22} + p^{12} T^{24} \) |
| 37 | \( ( 1 - 162 T^{2} + 12663 T^{4} - 590924 T^{6} + 12663 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 - 9 T - 45 T^{2} + 324 T^{3} + 4149 T^{4} - 13995 T^{5} - 117794 T^{6} - 13995 p T^{7} + 4149 p^{2} T^{8} + 324 p^{3} T^{9} - 45 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 43 | \( 1 + 135 T^{2} + 9711 T^{4} + 379192 T^{6} + 4757445 T^{8} - 11864493 p T^{10} - 35394356730 T^{12} - 11864493 p^{3} T^{14} + 4757445 p^{4} T^{16} + 379192 p^{6} T^{18} + 9711 p^{8} T^{20} + 135 p^{10} T^{22} + p^{12} T^{24} \) |
| 47 | \( 1 + 15 T^{2} - 4209 T^{4} + 13208 T^{6} + 9465885 T^{8} - 102475431 T^{10} - 22258948170 T^{12} - 102475431 p^{2} T^{14} + 9465885 p^{4} T^{16} + 13208 p^{6} T^{18} - 4209 p^{8} T^{20} + 15 p^{10} T^{22} + p^{12} T^{24} \) |
| 53 | \( ( 1 - 138 T^{2} + 5703 T^{4} - 126556 T^{6} + 5703 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( 1 + 207 T^{2} + 24615 T^{4} + 1533424 T^{6} + 37061253 T^{8} - 3338018991 T^{10} - 336678346794 T^{12} - 3338018991 p^{2} T^{14} + 37061253 p^{4} T^{16} + 1533424 p^{6} T^{18} + 24615 p^{8} T^{20} + 207 p^{10} T^{22} + p^{12} T^{24} \) |
| 61 | \( ( 1 + 9 T + 147 T^{2} + 1080 T^{3} + 10725 T^{4} + 112311 T^{5} + 816046 T^{6} + 112311 p T^{7} + 10725 p^{2} T^{8} + 1080 p^{3} T^{9} + 147 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 67 | \( 1 + 171 T^{2} + 15723 T^{4} + 593716 T^{6} - 24823179 T^{8} - 5696476335 T^{10} - 490863060498 T^{12} - 5696476335 p^{2} T^{14} - 24823179 p^{4} T^{16} + 593716 p^{6} T^{18} + 15723 p^{8} T^{20} + 171 p^{10} T^{22} + p^{12} T^{24} \) |
| 71 | \( ( 1 + 138 T^{2} + 13695 T^{4} + 1144348 T^{6} + 13695 p^{2} T^{8} + 138 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 + 135 T^{2} - 164 T^{3} + 135 p T^{4} + p^{3} T^{6} )^{4} \) |
| 79 | \( 1 - 381 T^{2} + 80043 T^{4} - 11776172 T^{6} + 1351016253 T^{8} - 129529123479 T^{10} + 10837987736766 T^{12} - 129529123479 p^{2} T^{14} + 1351016253 p^{4} T^{16} - 11776172 p^{6} T^{18} + 80043 p^{8} T^{20} - 381 p^{10} T^{22} + p^{12} T^{24} \) |
| 83 | \( 1 + 327 T^{2} + 57207 T^{4} + 6488176 T^{6} + 532197333 T^{8} + 34415866233 T^{10} + 2415401371734 T^{12} + 34415866233 p^{2} T^{14} + 532197333 p^{4} T^{16} + 6488176 p^{6} T^{18} + 57207 p^{8} T^{20} + 327 p^{10} T^{22} + p^{12} T^{24} \) |
| 89 | \( ( 1 + 12 T + 231 T^{2} + 2028 T^{3} + 231 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 97 | \( ( 1 + 3 T - 69 T^{2} + 1444 T^{3} + 333 T^{4} - 64359 T^{5} + 1516542 T^{6} - 64359 p T^{7} + 333 p^{2} T^{8} + 1444 p^{3} T^{9} - 69 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.45770374209748337553859942620, −3.31693674702775326566748339525, −3.16530322316889296550176530008, −3.14443462230379718564202655180, −3.08948982871837064003996642520, −2.88292870468792891226367333749, −2.72001193777535559057498439028, −2.57053121147095429392307065330, −2.55975494380091523602067502104, −2.52189152128740936633331913818, −2.45173885693077829670103588182, −2.26523516830889637338444696514, −2.11085470302951126007945266993, −2.05013313140130541316989733299, −2.03434019370722524547757883256, −1.88404234059583007923103572904, −1.59040847755227547342418417365, −1.53065952057537397413287902193, −1.49383052984470505889186830968, −1.48090022234490239944390635224, −1.45758119361943598645389012692, −1.21727292637397835746882709753, −0.994793888163488085960880372892, −0.52271847770359384192678164231, −0.27148536534579518737965646508,
0.27148536534579518737965646508, 0.52271847770359384192678164231, 0.994793888163488085960880372892, 1.21727292637397835746882709753, 1.45758119361943598645389012692, 1.48090022234490239944390635224, 1.49383052984470505889186830968, 1.53065952057537397413287902193, 1.59040847755227547342418417365, 1.88404234059583007923103572904, 2.03434019370722524547757883256, 2.05013313140130541316989733299, 2.11085470302951126007945266993, 2.26523516830889637338444696514, 2.45173885693077829670103588182, 2.52189152128740936633331913818, 2.55975494380091523602067502104, 2.57053121147095429392307065330, 2.72001193777535559057498439028, 2.88292870468792891226367333749, 3.08948982871837064003996642520, 3.14443462230379718564202655180, 3.16530322316889296550176530008, 3.31693674702775326566748339525, 3.45770374209748337553859942620
Plot not available for L-functions of degree greater than 10.
