| L(s) = 1 | − 2-s − 2·5-s + 4·7-s + 8-s + 2·10-s + 11-s + 6·13-s − 4·14-s − 16-s + 4·17-s + 8·19-s − 22-s − 4·23-s + 5·25-s − 6·26-s − 6·29-s − 4·34-s − 8·35-s + 12·37-s − 8·38-s − 2·40-s + 6·41-s − 4·43-s + 4·46-s + 12·47-s + 7·49-s − 5·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.894·5-s + 1.51·7-s + 0.353·8-s + 0.632·10-s + 0.301·11-s + 1.66·13-s − 1.06·14-s − 1/4·16-s + 0.970·17-s + 1.83·19-s − 0.213·22-s − 0.834·23-s + 25-s − 1.17·26-s − 1.11·29-s − 0.685·34-s − 1.35·35-s + 1.97·37-s − 1.29·38-s − 0.316·40-s + 0.937·41-s − 0.609·43-s + 0.589·46-s + 1.75·47-s + 49-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.159694886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.159694886\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.343205206338520900407985765824, −9.016091503447965786795765476800, −8.662491175949813007331539266276, −8.325251848935204919982984448834, −7.75383918100423541242416759766, −7.70165859237757295608375076862, −7.45892180871969647440146292828, −6.96137039359423436323101648569, −6.14359628000179431426173367395, −5.87026927332905148122852790369, −5.52683411646643825956499354129, −4.98084963502344651655389422168, −4.37223675632134724812614038870, −4.17729392851415610741626357656, −3.61561688605682570263444616910, −3.19444863872429356283236453158, −2.51016141691154590015405160160, −1.61974026695972400219071448857, −1.21137917161860257271172838558, −0.76445687512027243534363527773,
0.76445687512027243534363527773, 1.21137917161860257271172838558, 1.61974026695972400219071448857, 2.51016141691154590015405160160, 3.19444863872429356283236453158, 3.61561688605682570263444616910, 4.17729392851415610741626357656, 4.37223675632134724812614038870, 4.98084963502344651655389422168, 5.52683411646643825956499354129, 5.87026927332905148122852790369, 6.14359628000179431426173367395, 6.96137039359423436323101648569, 7.45892180871969647440146292828, 7.70165859237757295608375076862, 7.75383918100423541242416759766, 8.325251848935204919982984448834, 8.662491175949813007331539266276, 9.016091503447965786795765476800, 9.343205206338520900407985765824
