| L(s) = 1 | − 6·3-s + 16·4-s + 27·9-s − 96·12-s + 192·16-s + 156·17-s + 192·23-s + 106·25-s − 108·27-s + 36·29-s + 432·36-s − 1.04e3·43-s − 1.15e3·48-s + 682·49-s − 936·51-s + 1.11e3·53-s + 148·61-s + 2.04e3·64-s + 2.49e3·68-s − 1.15e3·69-s − 636·75-s + 1.40e3·79-s + 405·81-s − 216·87-s + 3.07e3·92-s + 1.69e3·100-s + 1.98e3·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2·4-s + 9-s − 2.30·12-s + 3·16-s + 2.22·17-s + 1.74·23-s + 0.847·25-s − 0.769·27-s + 0.230·29-s + 2·36-s − 3.71·43-s − 3.46·48-s + 1.98·49-s − 2.56·51-s + 2.89·53-s + 0.310·61-s + 4·64-s + 4.45·68-s − 2.00·69-s − 0.979·75-s + 2.00·79-s + 5/9·81-s − 0.266·87-s + 3.48·92-s + 1.69·100-s + 1.95·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.903290034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.903290034\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 106 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 682 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 1366 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 78 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 8242 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 96 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 13786 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 19510 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9614 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 524 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 117646 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 558 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 78982 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 600082 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 507886 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 312910 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 704 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 355030 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 369538 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1813246 T^{2} + p^{6} T^{4} \) |
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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
−10.73553956874182388077062748758, −10.31709816777718576249370920241, −10.10697745563384368033647550658, −9.824834403123852412048788783493, −8.760890457466282369360590478108, −8.526892998998024443091679029037, −7.58446194570514473587095502389, −7.54196263282179002308020102137, −6.85770087983768156585641910833, −6.77493939022990281976704007702, −6.11274010954071403906120831092, −5.65954133900115288703694487464, −5.19105321088377414643194425256, −4.92616042861740135160350351223, −3.62383725782463126359397899111, −3.42044431367593246000085483793, −2.69597295464047037580463171420, −1.98206246669603307863181702158, −1.06986029452229188950017688255, −0.922803367563866036483708166567,
0.922803367563866036483708166567, 1.06986029452229188950017688255, 1.98206246669603307863181702158, 2.69597295464047037580463171420, 3.42044431367593246000085483793, 3.62383725782463126359397899111, 4.92616042861740135160350351223, 5.19105321088377414643194425256, 5.65954133900115288703694487464, 6.11274010954071403906120831092, 6.77493939022990281976704007702, 6.85770087983768156585641910833, 7.54196263282179002308020102137, 7.58446194570514473587095502389, 8.526892998998024443091679029037, 8.760890457466282369360590478108, 9.824834403123852412048788783493, 10.10697745563384368033647550658, 10.31709816777718576249370920241, 10.73553956874182388077062748758
