| L(s) = 1 | − 12·9-s − 96·25-s − 96·29-s − 192·41-s − 80·53-s − 192·61-s − 576·73-s + 90·81-s − 192·89-s − 384·97-s − 960·101-s − 192·109-s + 112·113-s + 584·121-s − 192·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 384·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 3.83·25-s − 3.31·29-s − 4.68·41-s − 1.50·53-s − 3.14·61-s − 7.89·73-s + 10/9·81-s − 2.15·89-s − 3.95·97-s − 9.50·101-s − 1.76·109-s + 0.991·113-s + 4.82·121-s − 1.53·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.27·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.845382472\times10^{-6}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.845382472\times10^{-6}\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T^{2} )^{4} \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 48 T^{2} + 96 T^{3} + 1346 T^{4} + 96 p^{2} T^{5} + 48 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 - 584 T^{2} + 167452 T^{4} - 31553912 T^{6} + 4373265670 T^{8} - 31553912 p^{4} T^{10} + 167452 p^{8} T^{12} - 584 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 + 192 T^{2} - 1344 T^{3} + 7970 T^{4} - 1344 p^{2} T^{5} + 192 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 528 T^{2} - 672 T^{3} + 220130 T^{4} - 672 p^{2} T^{5} + 528 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( 1 - 1928 T^{2} + 1783900 T^{4} - 1059171896 T^{6} + 447269664262 T^{8} - 1059171896 p^{4} T^{10} + 1783900 p^{8} T^{12} - 1928 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( 1 - 1928 T^{2} + 1932124 T^{4} - 1454834360 T^{6} + 872243294278 T^{8} - 1454834360 p^{4} T^{10} + 1932124 p^{8} T^{12} - 1928 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 + 48 T + 3556 T^{2} + 114192 T^{3} + 4574694 T^{4} + 114192 p^{2} T^{5} + 3556 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 4232 T^{2} + 9689116 T^{4} - 14891091896 T^{6} + 16606329017926 T^{8} - 14891091896 p^{4} T^{10} + 9689116 p^{8} T^{12} - 4232 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 + 1636 T^{2} - 105984 T^{3} + 99750 T^{4} - 105984 p^{2} T^{5} + 1636 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 96 T + 7536 T^{2} + 422016 T^{3} + 19400354 T^{4} + 422016 p^{2} T^{5} + 7536 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 - 5576 T^{2} + 20016988 T^{4} - 46850763512 T^{6} + 96062127228550 T^{8} - 46850763512 p^{4} T^{10} + 20016988 p^{8} T^{12} - 5576 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 - 8456 T^{2} + 28082716 T^{4} - 44170889528 T^{6} + 55756301244742 T^{8} - 44170889528 p^{4} T^{10} + 28082716 p^{8} T^{12} - 8456 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 40 T + 7804 T^{2} + 343384 T^{3} + 29078758 T^{4} + 343384 p^{2} T^{5} + 7804 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 21896 T^{2} + 224587228 T^{4} - 1409309727032 T^{6} + 5922457985328262 T^{8} - 1409309727032 p^{4} T^{10} + 224587228 p^{8} T^{12} - 21896 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 96 T + 17184 T^{2} + 1068960 T^{3} + 99806114 T^{4} + 1068960 p^{2} T^{5} + 17184 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 23240 T^{2} + 238231516 T^{4} - 1505735752952 T^{6} + 7309602668695558 T^{8} - 1505735752952 p^{4} T^{10} + 238231516 p^{8} T^{12} - 23240 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( 1 - 17288 T^{2} + 123269212 T^{4} - 392735012024 T^{6} + 943741770326854 T^{8} - 392735012024 p^{4} T^{10} + 123269212 p^{8} T^{12} - 17288 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 288 T + 49248 T^{2} + 5605536 T^{3} + 476605250 T^{4} + 5605536 p^{2} T^{5} + 49248 p^{4} T^{6} + 288 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 19976 T^{2} + 232744732 T^{4} - 1840641279032 T^{6} + 12411054418089286 T^{8} - 1840641279032 p^{4} T^{10} + 232744732 p^{8} T^{12} - 19976 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 - 37256 T^{2} + 681767260 T^{4} - 7971195138104 T^{6} + 65017943250589702 T^{8} - 7971195138104 p^{4} T^{10} + 681767260 p^{8} T^{12} - 37256 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 96 T + 33456 T^{2} + 2237760 T^{3} + 403501346 T^{4} + 2237760 p^{2} T^{5} + 33456 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 192 T + 30432 T^{2} + 2692416 T^{3} + 3014210 p T^{4} + 2692416 p^{2} T^{5} + 30432 p^{4} T^{6} + 192 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.51767114345046814929099973866, −3.51047532431995785382970996223, −3.43555818434253526542217159692, −3.18293428422099513594451532255, −3.12757252164347698276755103553, −2.91158212140529711910226492502, −2.83892678606563520540463816213, −2.56057800192831619664609694011, −2.53896155499263656777255436367, −2.46128244893881523623788075565, −2.35905800443027980527073379666, −2.22962590835577791288455537833, −2.13719768330973553346349116641, −1.68167616497944639248359253051, −1.64020090397032826157084842923, −1.46985615752627608464890274053, −1.44412024743802680324948925187, −1.36305899031019637348130426512, −1.34485607441094577944765625465, −1.31911325804793767938378606198, −0.953227703699964225578326892809, −0.15353258736914831458029573307, −0.087332165677424524636306802656, −0.04631352004938289029683868666, −0.01629745578550010714639445785,
0.01629745578550010714639445785, 0.04631352004938289029683868666, 0.087332165677424524636306802656, 0.15353258736914831458029573307, 0.953227703699964225578326892809, 1.31911325804793767938378606198, 1.34485607441094577944765625465, 1.36305899031019637348130426512, 1.44412024743802680324948925187, 1.46985615752627608464890274053, 1.64020090397032826157084842923, 1.68167616497944639248359253051, 2.13719768330973553346349116641, 2.22962590835577791288455537833, 2.35905800443027980527073379666, 2.46128244893881523623788075565, 2.53896155499263656777255436367, 2.56057800192831619664609694011, 2.83892678606563520540463816213, 2.91158212140529711910226492502, 3.12757252164347698276755103553, 3.18293428422099513594451532255, 3.43555818434253526542217159692, 3.51047532431995785382970996223, 3.51767114345046814929099973866
Plot not available for L-functions of degree greater than 10.
