| L(s) = 1 | − 4-s − 2·5-s − 12·11-s + 16-s − 8·19-s + 2·20-s − 25-s + 4·29-s − 4·31-s − 12·41-s + 12·44-s − 49-s + 24·55-s − 8·59-s − 24·61-s − 64-s − 24·71-s + 8·76-s + 32·79-s − 2·80-s + 28·89-s + 16·95-s + 100-s − 12·101-s − 28·109-s − 4·116-s + 86·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 3.61·11-s + 1/4·16-s − 1.83·19-s + 0.447·20-s − 1/5·25-s + 0.742·29-s − 0.718·31-s − 1.87·41-s + 1.80·44-s − 1/7·49-s + 3.23·55-s − 1.04·59-s − 3.07·61-s − 1/8·64-s − 2.84·71-s + 0.917·76-s + 3.60·79-s − 0.223·80-s + 2.96·89-s + 1.64·95-s + 1/10·100-s − 1.19·101-s − 2.68·109-s − 0.371·116-s + 7.81·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−10.55657875152966429594158260903, −10.22779761165845466921252137232, −9.447318067158642027712950685219, −9.060388869657719106444782658564, −8.407880945464455177552508213826, −8.162158020430392979246648358149, −7.70957274759391002765155104293, −7.67206995858947566629658740207, −6.90811881186133900093805027418, −6.22567570122555689334523103263, −5.81983847326637871828440474152, −5.05665323546398163036995543833, −4.90870091823457769047657154342, −4.45817607095627304265987613552, −3.66733773100512465290388737142, −3.08022572185866180901814861037, −2.56552214296133310771372601527, −1.86453934139038829702884379634, 0, 0,
1.86453934139038829702884379634, 2.56552214296133310771372601527, 3.08022572185866180901814861037, 3.66733773100512465290388737142, 4.45817607095627304265987613552, 4.90870091823457769047657154342, 5.05665323546398163036995543833, 5.81983847326637871828440474152, 6.22567570122555689334523103263, 6.90811881186133900093805027418, 7.67206995858947566629658740207, 7.70957274759391002765155104293, 8.162158020430392979246648358149, 8.407880945464455177552508213826, 9.060388869657719106444782658564, 9.447318067158642027712950685219, 10.22779761165845466921252137232, 10.55657875152966429594158260903
