| L(s) = 1 | − 12·3-s − 148·7-s + 63·9-s − 32·13-s − 748·19-s + 1.77e3·21-s − 125·25-s + 216·27-s + 2.85e3·31-s + 544·37-s + 384·39-s + 3.87e3·43-s + 1.16e4·49-s + 8.97e3·57-s + 4.46e3·61-s − 9.32e3·63-s + 1.08e4·67-s − 1.07e3·73-s + 1.50e3·75-s − 1.99e4·79-s − 7.69e3·81-s + 4.73e3·91-s − 3.42e4·93-s − 2.14e4·97-s + 1.84e4·103-s − 1.10e4·109-s − 6.52e3·111-s + ⋯ |
| L(s) = 1 | − 4/3·3-s − 3.02·7-s + 7/9·9-s − 0.189·13-s − 2.07·19-s + 4.02·21-s − 1/5·25-s + 8/27·27-s + 2.96·31-s + 0.397·37-s + 0.252·39-s + 2.09·43-s + 4.84·49-s + 2.76·57-s + 1.20·61-s − 2.34·63-s + 2.41·67-s − 0.201·73-s + 4/15·75-s − 3.19·79-s − 1.17·81-s + 0.571·91-s − 3.95·93-s − 2.27·97-s + 1.74·103-s − 0.926·109-s − 0.529·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3388380184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3388380184\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 4 p T + p^{4} T^{2} \) |
| 5 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 74 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 14702 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 16 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 136622 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 374 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 127502 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 495058 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 46 p T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 272 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5046002 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 1936 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 522418 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 14329618 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 1036142 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2234 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5416 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 48401282 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 538 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 9962 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 73243022 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 97824962 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10726 T + p^{4} T^{2} )^{2} \) |
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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
−11.73874526512085778621334122009, −11.23763782833837738010019683746, −10.70706006958474586585770521211, −10.16733361267571196373815362993, −9.827414809871944492672170072820, −9.713731334290725988661201095547, −8.677206695230065220899860684887, −8.583334482283566531609249309196, −7.52549854493478648266251775864, −6.84614031760951132868145287591, −6.49278993771628854706926269031, −6.20505267740843815531721546420, −5.90344102554362219430606126064, −5.06095700379983654665490424472, −4.18266674192489104079082621507, −3.91126481734179611179582025498, −2.69086459292330732863351691475, −2.66992672939150750552558085832, −0.904196334665718344359970176596, −0.27619587478908707681810448685,
0.27619587478908707681810448685, 0.904196334665718344359970176596, 2.66992672939150750552558085832, 2.69086459292330732863351691475, 3.91126481734179611179582025498, 4.18266674192489104079082621507, 5.06095700379983654665490424472, 5.90344102554362219430606126064, 6.20505267740843815531721546420, 6.49278993771628854706926269031, 6.84614031760951132868145287591, 7.52549854493478648266251775864, 8.583334482283566531609249309196, 8.677206695230065220899860684887, 9.713731334290725988661201095547, 9.827414809871944492672170072820, 10.16733361267571196373815362993, 10.70706006958474586585770521211, 11.23763782833837738010019683746, 11.73874526512085778621334122009
