| L(s) = 1 | + 1.58·2-s − 2.51·3-s + 0.517·4-s − 3.98·6-s + 3.57·7-s − 2.35·8-s + 3.31·9-s − 5.69·11-s − 1.30·12-s − 3.97·13-s + 5.66·14-s − 4.76·16-s + 1.07·17-s + 5.26·18-s + 4.17·19-s − 8.97·21-s − 9.03·22-s − 4.76·23-s + 5.91·24-s − 6.30·26-s − 0.803·27-s + 1.84·28-s − 3.94·29-s + 1.79·31-s − 2.85·32-s + 14.3·33-s + 1.70·34-s + ⋯ |
| L(s) = 1 | + 1.12·2-s − 1.45·3-s + 0.258·4-s − 1.62·6-s + 1.34·7-s − 0.831·8-s + 1.10·9-s − 1.71·11-s − 0.375·12-s − 1.10·13-s + 1.51·14-s − 1.19·16-s + 0.260·17-s + 1.24·18-s + 0.957·19-s − 1.95·21-s − 1.92·22-s − 0.992·23-s + 1.20·24-s − 1.23·26-s − 0.154·27-s + 0.349·28-s − 0.731·29-s + 0.322·31-s − 0.505·32-s + 2.49·33-s + 0.292·34-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
| \(L(1)\) |
\(\approx\) |
\(1.223323448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223323448\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.58T + 2T^{2} \) |
| 3 | \( 1 + 2.51T + 3T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 11 | \( 1 + 5.69T + 11T^{2} \) |
| 13 | \( 1 + 3.97T + 13T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 + 4.76T + 23T^{2} \) |
| 29 | \( 1 + 3.94T + 29T^{2} \) |
| 31 | \( 1 - 1.79T + 31T^{2} \) |
| 37 | \( 1 + 5.75T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 6.07T + 67T^{2} \) |
| 71 | \( 1 - 9.91T + 71T^{2} \) |
| 73 | \( 1 + 2.95T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 4.09T + 83T^{2} \) |
| 89 | \( 1 - 9.20T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−7.74748096378111486068920424973, −7.35077744946777104536432109858, −6.30133636275286555279124399819, −5.39717426147137932887301724338, −5.32872079185569450824624356546, −4.81007343118905481328017820724, −4.03014147831555358205176213684, −2.86701029352423834901444124394, −1.98564043202973866677627032818, −0.50829274243733967196836102018,
0.50829274243733967196836102018, 1.98564043202973866677627032818, 2.86701029352423834901444124394, 4.03014147831555358205176213684, 4.81007343118905481328017820724, 5.32872079185569450824624356546, 5.39717426147137932887301724338, 6.30133636275286555279124399819, 7.35077744946777104536432109858, 7.74748096378111486068920424973
