| L(s) = 1 | − 3·4-s − 2·5-s − 4·7-s − 6·9-s + 4·13-s + 5·16-s + 6·20-s + 4·23-s + 3·25-s + 12·28-s − 29-s + 8·35-s + 18·36-s + 12·45-s − 2·49-s − 12·52-s − 12·53-s − 16·59-s + 24·63-s − 3·64-s − 8·65-s + 4·67-s − 24·71-s − 10·80-s + 27·81-s − 28·83-s − 16·91-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 0.894·5-s − 1.51·7-s − 2·9-s + 1.10·13-s + 5/4·16-s + 1.34·20-s + 0.834·23-s + 3/5·25-s + 2.26·28-s − 0.185·29-s + 1.35·35-s + 3·36-s + 1.78·45-s − 2/7·49-s − 1.66·52-s − 1.64·53-s − 2.08·59-s + 3.02·63-s − 3/8·64-s − 0.992·65-s + 0.488·67-s − 2.84·71-s − 1.11·80-s + 3·81-s − 3.07·83-s − 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 609725 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 609725 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 29 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.239574112295842649010973903379, −7.64947192507391358342580089136, −7.05705578137635354910935495229, −6.44692364751045357757742318687, −6.01956800466047829467852819351, −5.76866201806355274035684105479, −5.07082360411438799021122699455, −4.64971827708119930945851978376, −4.02824177002899588849770856217, −3.52751833946552145657108459336, −2.96744633894375249844428852217, −2.95203182599403093553876063934, −1.30560375542689889446851628523, 0, 0,
1.30560375542689889446851628523, 2.95203182599403093553876063934, 2.96744633894375249844428852217, 3.52751833946552145657108459336, 4.02824177002899588849770856217, 4.64971827708119930945851978376, 5.07082360411438799021122699455, 5.76866201806355274035684105479, 6.01956800466047829467852819351, 6.44692364751045357757742318687, 7.05705578137635354910935495229, 7.64947192507391358342580089136, 8.239574112295842649010973903379
