| L(s) = 1 | − 6·3-s − 50·5-s − 140·7-s + 66·11-s + 338·13-s + 300·15-s − 1.87e3·17-s + 2.13e3·19-s + 840·21-s + 3.15e3·23-s + 1.87e3·25-s − 1.24e3·27-s − 5.10e3·29-s − 5.48e3·31-s − 396·33-s + 7.00e3·35-s − 2.03e4·37-s − 2.02e3·39-s + 1.37e4·41-s − 2.46e3·43-s + 1.09e4·47-s − 1.56e4·49-s + 1.12e4·51-s − 3.60e3·53-s − 3.30e3·55-s − 1.28e4·57-s − 6.84e3·59-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 0.894·5-s − 1.07·7-s + 0.164·11-s + 0.554·13-s + 0.344·15-s − 1.57·17-s + 1.35·19-s + 0.415·21-s + 1.24·23-s + 3/5·25-s − 0.327·27-s − 1.12·29-s − 1.02·31-s − 0.0633·33-s + 0.965·35-s − 2.43·37-s − 0.213·39-s + 1.27·41-s − 0.203·43-s + 0.720·47-s − 0.931·49-s + 0.604·51-s − 0.176·53-s − 0.147·55-s − 0.521·57-s − 0.256·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 p T + 4 p^{2} T^{2} + 2 p^{6} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 20 p T + 35250 T^{2} + 20 p^{6} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 p T + 260716 T^{2} - 6 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1872 T + 3642166 T^{2} + 1872 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2134 T + 4896828 T^{2} - 2134 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3150 T + 15344692 T^{2} - 3150 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5100 T + 30486514 T^{2} + 5100 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5486 T + 1699572 p T^{2} + 5486 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 20312 T + 241097850 T^{2} + 20312 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 336 p T + 246025510 T^{2} - 336 p^{6} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2462 T + 228669972 T^{2} + 2462 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10908 T + 469083874 T^{2} - 10908 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3600 T + 487941310 T^{2} + 3600 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6846 T + 1418876188 T^{2} + 6846 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 7492 T + 1654044594 T^{2} - 7492 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3944 T + 2625468234 T^{2} + 3944 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 62178 T + 4503851812 T^{2} + 62178 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 15688 T + 2908074018 T^{2} - 15688 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 71084 T + 5816843958 T^{2} + 71084 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 127836 T + 10546098826 T^{2} + 127836 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 30420 T + 5621431462 T^{2} - 30420 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11908 T + 7953896646 T^{2} - 11908 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.85296693502523160486050076321, −10.69828338444247772078740620347, −9.894640493068922976554189971410, −9.384083706393943590309738564159, −8.894260499773064918094405094110, −8.743323462925306358429535825827, −7.83568200970256301956505283255, −7.27900171127076813959405234914, −6.95179327770621060794936698375, −6.55078649681062159071479488400, −5.66051423753789262912481691991, −5.45368693608211071033158355040, −4.54114077175766854345157445654, −4.04966537033901366728871229558, −3.28558163293587266715229605293, −3.08267210555065790256403055001, −1.92230236672168376032784401895, −1.11541083437763046937381973625, 0, 0,
1.11541083437763046937381973625, 1.92230236672168376032784401895, 3.08267210555065790256403055001, 3.28558163293587266715229605293, 4.04966537033901366728871229558, 4.54114077175766854345157445654, 5.45368693608211071033158355040, 5.66051423753789262912481691991, 6.55078649681062159071479488400, 6.95179327770621060794936698375, 7.27900171127076813959405234914, 7.83568200970256301956505283255, 8.743323462925306358429535825827, 8.894260499773064918094405094110, 9.384083706393943590309738564159, 9.894640493068922976554189971410, 10.69828338444247772078740620347, 10.85296693502523160486050076321
