| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 9-s + 4·13-s + 5·16-s − 17-s − 2·18-s − 8·19-s − 10·25-s − 8·26-s − 6·32-s + 2·34-s + 3·36-s + 16·38-s − 8·43-s + 24·47-s − 10·49-s + 20·50-s + 12·52-s + 12·53-s − 24·59-s + 7·64-s − 8·67-s − 3·68-s − 4·72-s − 24·76-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 1/3·9-s + 1.10·13-s + 5/4·16-s − 0.242·17-s − 0.471·18-s − 1.83·19-s − 2·25-s − 1.56·26-s − 1.06·32-s + 0.342·34-s + 1/2·36-s + 2.59·38-s − 1.21·43-s + 3.50·47-s − 1.42·49-s + 2.82·50-s + 1.66·52-s + 1.64·53-s − 3.12·59-s + 7/8·64-s − 0.977·67-s − 0.363·68-s − 0.471·72-s − 2.75·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176868 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176868 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.943787248918964717870933062631, −8.367508603382134314146293072792, −8.268041625641194749866153864112, −7.50641789037179894281138679970, −7.25021044600316149163291697385, −6.53161459670871116875221212820, −5.97162555708025444414279518144, −5.96956680213085492365415051586, −4.88852544958904667852187403353, −4.05372696693836488551925909574, −3.77794172579737779281492402581, −2.73027674275297291623131819050, −2.04126125287275460099535053657, −1.38191572633065703837771005303, 0,
1.38191572633065703837771005303, 2.04126125287275460099535053657, 2.73027674275297291623131819050, 3.77794172579737779281492402581, 4.05372696693836488551925909574, 4.88852544958904667852187403353, 5.96956680213085492365415051586, 5.97162555708025444414279518144, 6.53161459670871116875221212820, 7.25021044600316149163291697385, 7.50641789037179894281138679970, 8.268041625641194749866153864112, 8.367508603382134314146293072792, 8.943787248918964717870933062631
