| L(s) = 1 | + 2-s + 4·5-s − 8-s + 3·9-s + 4·10-s + 11-s + 4·13-s − 16-s + 4·17-s + 3·18-s + 6·19-s + 22-s − 4·23-s + 5·25-s + 4·26-s − 4·29-s + 2·31-s + 4·34-s − 10·37-s + 6·38-s − 4·40-s + 8·41-s − 16·43-s + 12·45-s − 4·46-s − 2·47-s + 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.78·5-s − 0.353·8-s + 9-s + 1.26·10-s + 0.301·11-s + 1.10·13-s − 1/4·16-s + 0.970·17-s + 0.707·18-s + 1.37·19-s + 0.213·22-s − 0.834·23-s + 25-s + 0.784·26-s − 0.742·29-s + 0.359·31-s + 0.685·34-s − 1.64·37-s + 0.973·38-s − 0.632·40-s + 1.24·41-s − 2.43·43-s + 1.78·45-s − 0.589·46-s − 0.291·47-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.377827970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.377827970\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−10.14135246204694752895246302620, −9.728487503692433845343612539992, −9.533770316473619907996675059074, −8.945605507371630456180816962561, −8.296669659523810594535219946907, −8.278653977682538368213433648929, −7.43508509824019733371962286826, −6.87389443856581219564747414437, −6.83697247476284849727067962289, −5.92343046172389565825877820298, −5.92148293867290485265747582528, −5.35072494980282081715338562000, −5.17864934667485903209640128342, −4.34838555163970661450082339628, −3.94538224276158265367873263286, −3.35340507671052236674984709731, −3.01498951779872565081724709347, −1.91472600607395798013754341360, −1.75371800366853408060703037084, −1.01892137604708122341118930824,
1.01892137604708122341118930824, 1.75371800366853408060703037084, 1.91472600607395798013754341360, 3.01498951779872565081724709347, 3.35340507671052236674984709731, 3.94538224276158265367873263286, 4.34838555163970661450082339628, 5.17864934667485903209640128342, 5.35072494980282081715338562000, 5.92148293867290485265747582528, 5.92343046172389565825877820298, 6.83697247476284849727067962289, 6.87389443856581219564747414437, 7.43508509824019733371962286826, 8.278653977682538368213433648929, 8.296669659523810594535219946907, 8.945605507371630456180816962561, 9.533770316473619907996675059074, 9.728487503692433845343612539992, 10.14135246204694752895246302620
