| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 3·5-s − 4·6-s + 4·8-s + 3·9-s − 6·10-s + 11-s − 6·12-s − 2·13-s + 6·15-s + 5·16-s + 17-s + 6·18-s − 3·19-s − 9·20-s + 2·22-s − 7·23-s − 8·24-s + 25-s − 4·26-s − 4·27-s + 29-s + 12·30-s + 4·31-s + 6·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.34·5-s − 1.63·6-s + 1.41·8-s + 9-s − 1.89·10-s + 0.301·11-s − 1.73·12-s − 0.554·13-s + 1.54·15-s + 5/4·16-s + 0.242·17-s + 1.41·18-s − 0.688·19-s − 2.01·20-s + 0.426·22-s − 1.45·23-s − 1.63·24-s + 1/5·25-s − 0.784·26-s − 0.769·27-s + 0.185·29-s + 2.19·30-s + 0.718·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 17 T + 154 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 144 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 26 T + 318 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.188849464902300460548092872658, −7.83442632807584960067983105142, −7.42590511407383774850594312249, −6.90282458555615350675509397390, −6.72209315384959419675421368395, −6.62979300972915581834554219602, −5.94786860717645271088139912042, −5.58704493595776581039852349174, −5.19446594025614260546394525330, −5.04987953835157913215592147057, −4.32645941060827123763967334007, −4.23527472924228621408260472248, −3.73636829008690494363744241843, −3.59844997507620280971808870077, −2.86985591270793636190150429916, −2.39166072025164051913154222172, −1.68874239473002760244411775936, −1.31591471145023310684852207232, 0, 0,
1.31591471145023310684852207232, 1.68874239473002760244411775936, 2.39166072025164051913154222172, 2.86985591270793636190150429916, 3.59844997507620280971808870077, 3.73636829008690494363744241843, 4.23527472924228621408260472248, 4.32645941060827123763967334007, 5.04987953835157913215592147057, 5.19446594025614260546394525330, 5.58704493595776581039852349174, 5.94786860717645271088139912042, 6.62979300972915581834554219602, 6.72209315384959419675421368395, 6.90282458555615350675509397390, 7.42590511407383774850594312249, 7.83442632807584960067983105142, 8.188849464902300460548092872658
