| L(s) = 1 | − 0.414·2-s − 1.82·4-s − 5-s − 4.82·7-s + 1.58·8-s + 0.414·10-s − 0.828·11-s − 2·13-s + 1.99·14-s + 3·16-s − 2.82·17-s − 4.82·19-s + 1.82·20-s + 0.343·22-s + 3.17·23-s + 25-s + 0.828·26-s + 8.82·28-s − 29-s + 6.48·31-s − 4.41·32-s + 1.17·34-s + 4.82·35-s − 8.48·37-s + 1.99·38-s − 1.58·40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.914·4-s − 0.447·5-s − 1.82·7-s + 0.560·8-s + 0.130·10-s − 0.249·11-s − 0.554·13-s + 0.534·14-s + 0.750·16-s − 0.685·17-s − 1.10·19-s + 0.408·20-s + 0.0731·22-s + 0.661·23-s + 0.200·25-s + 0.162·26-s + 1.66·28-s − 0.185·29-s + 1.16·31-s − 0.780·32-s + 0.200·34-s + 0.816·35-s − 1.39·37-s + 0.324·38-s − 0.250·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4462992790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4462992790\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 - 6.48T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 2.48T + 79T^{2} \) |
| 83 | \( 1 + 7.17T + 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−9.569735796860283104887902752783, −8.935221445141604486704813754802, −8.228094159143384664934506081671, −7.13012654704060293700489516664, −6.51560178460189214488763299907, −5.42031063076244516808654166364, −4.38427934121461106876595539207, −3.61131050761625910666601968286, −2.55173832700983347070171173196, −0.48854540853707181561886723604,
0.48854540853707181561886723604, 2.55173832700983347070171173196, 3.61131050761625910666601968286, 4.38427934121461106876595539207, 5.42031063076244516808654166364, 6.51560178460189214488763299907, 7.13012654704060293700489516664, 8.228094159143384664934506081671, 8.935221445141604486704813754802, 9.569735796860283104887902752783
