| L(s) = 1 | + 2-s − 4-s − 3·8-s − 3·9-s − 2·11-s + 4·13-s − 16-s − 3·18-s − 2·22-s + 8·23-s + 25-s + 4·26-s + 5·32-s + 3·36-s − 4·37-s + 2·44-s + 8·46-s − 24·47-s − 14·49-s + 50-s − 4·52-s + 8·59-s − 20·61-s + 7·64-s + 16·71-s + 9·72-s + 28·73-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 9-s − 0.603·11-s + 1.10·13-s − 1/4·16-s − 0.707·18-s − 0.426·22-s + 1.66·23-s + 1/5·25-s + 0.784·26-s + 0.883·32-s + 1/2·36-s − 0.657·37-s + 0.301·44-s + 1.17·46-s − 3.50·47-s − 2·49-s + 0.141·50-s − 0.554·52-s + 1.04·59-s − 2.56·61-s + 7/8·64-s + 1.89·71-s + 1.06·72-s + 3.27·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.248642768475844318802620430882, −8.227294201073412154999968421070, −7.52567387860361416690294259657, −6.73114079621071100714169312036, −6.32256355952646480381816699905, −6.16821463973303244938343944630, −5.19072755325840146238482378089, −5.09766059609940450216711100862, −4.80763834609260757451310676100, −3.82573290551710488813917015410, −3.25423179226253980082861279896, −3.19776287627196162761105647639, −2.27052010539287522961646416611, −1.21814035480499692293997625850, 0,
1.21814035480499692293997625850, 2.27052010539287522961646416611, 3.19776287627196162761105647639, 3.25423179226253980082861279896, 3.82573290551710488813917015410, 4.80763834609260757451310676100, 5.09766059609940450216711100862, 5.19072755325840146238482378089, 6.16821463973303244938343944630, 6.32256355952646480381816699905, 6.73114079621071100714169312036, 7.52567387860361416690294259657, 8.227294201073412154999968421070, 8.248642768475844318802620430882
