| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s + 2·7-s − 4·8-s + 2·9-s + 4·10-s − 4·13-s − 4·14-s + 5·16-s − 4·17-s − 4·18-s − 6·20-s + 3·25-s + 8·26-s + 6·28-s − 12·29-s − 6·32-s + 8·34-s − 4·35-s + 6·36-s + 12·37-s + 8·40-s − 4·41-s + 8·43-s − 4·45-s − 16·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.755·7-s − 1.41·8-s + 2/3·9-s + 1.26·10-s − 1.10·13-s − 1.06·14-s + 5/4·16-s − 0.970·17-s − 0.942·18-s − 1.34·20-s + 3/5·25-s + 1.56·26-s + 1.13·28-s − 2.22·29-s − 1.06·32-s + 1.37·34-s − 0.676·35-s + 36-s + 1.97·37-s + 1.26·40-s − 0.624·41-s + 1.21·43-s − 0.596·45-s − 2.33·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 214 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 190 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.69290706061046991419729170522, −7.64985001706633197573353688033, −7.12410196158094950650297801529, −6.62923271545596838305663194834, −6.41715937623756269132467792109, −6.23735449884101537305858375725, −5.36737590160804753915907662625, −5.27176126435502999331733538728, −4.86959097352514473556504623510, −4.33114404947085658146874566075, −4.07166116468635060645889135818, −3.78682559478378396781756919967, −3.01740286926770876689505843721, −2.82989658276470805211087713559, −2.25843032292322031240915289699, −1.82238133592809602959791585716, −1.50404889698204852610377958237, −0.913297055630509466733777954132, 0, 0,
0.913297055630509466733777954132, 1.50404889698204852610377958237, 1.82238133592809602959791585716, 2.25843032292322031240915289699, 2.82989658276470805211087713559, 3.01740286926770876689505843721, 3.78682559478378396781756919967, 4.07166116468635060645889135818, 4.33114404947085658146874566075, 4.86959097352514473556504623510, 5.27176126435502999331733538728, 5.36737590160804753915907662625, 6.23735449884101537305858375725, 6.41715937623756269132467792109, 6.62923271545596838305663194834, 7.12410196158094950650297801529, 7.64985001706633197573353688033, 7.69290706061046991419729170522
