| L(s) = 1 | − 1.93·2-s − 1.97·3-s + 1.75·4-s + 3.82·6-s − 0.0385·7-s + 0.470·8-s + 0.893·9-s − 2.40·11-s − 3.46·12-s + 2.78·13-s + 0.0747·14-s − 4.42·16-s + 6.12·17-s − 1.73·18-s + 0.288·19-s + 0.0760·21-s + 4.65·22-s + 4.83·23-s − 0.927·24-s − 5.39·26-s + 4.15·27-s − 0.0677·28-s + 3.96·29-s + 4.73·31-s + 7.63·32-s + 4.73·33-s − 11.8·34-s + ⋯ |
| L(s) = 1 | − 1.37·2-s − 1.13·3-s + 0.878·4-s + 1.56·6-s − 0.0145·7-s + 0.166·8-s + 0.297·9-s − 0.723·11-s − 1.00·12-s + 0.772·13-s + 0.0199·14-s − 1.10·16-s + 1.48·17-s − 0.408·18-s + 0.0661·19-s + 0.0165·21-s + 0.992·22-s + 1.00·23-s − 0.189·24-s − 1.05·26-s + 0.799·27-s − 0.0128·28-s + 0.735·29-s + 0.850·31-s + 1.35·32-s + 0.824·33-s − 2.03·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6270947606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6270947606\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 3 | \( 1 + 1.97T + 3T^{2} \) |
| 7 | \( 1 + 0.0385T + 7T^{2} \) |
| 11 | \( 1 + 2.40T + 11T^{2} \) |
| 13 | \( 1 - 2.78T + 13T^{2} \) |
| 17 | \( 1 - 6.12T + 17T^{2} \) |
| 19 | \( 1 - 0.288T + 19T^{2} \) |
| 23 | \( 1 - 4.83T + 23T^{2} \) |
| 29 | \( 1 - 3.96T + 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 - 0.878T + 37T^{2} \) |
| 41 | \( 1 + 2.13T + 41T^{2} \) |
| 43 | \( 1 - 5.90T + 43T^{2} \) |
| 47 | \( 1 - 3.01T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 - 1.33T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 1.78T + 83T^{2} \) |
| 89 | \( 1 + 0.156T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.190051440895637894194681246907, −7.52554386650656870917931270888, −6.70963904937253658818392957831, −6.13244038412229941088119558565, −5.24984915350951040543495607352, −4.78119126177663913632978947864, −3.52450036826295663934377513347, −2.53502254099531284059279093058, −1.21156204195388829758447990232, −0.65441073608401725392437197850,
0.65441073608401725392437197850, 1.21156204195388829758447990232, 2.53502254099531284059279093058, 3.52450036826295663934377513347, 4.78119126177663913632978947864, 5.24984915350951040543495607352, 6.13244038412229941088119558565, 6.70963904937253658818392957831, 7.52554386650656870917931270888, 8.190051440895637894194681246907
