| L(s) = 1 | + 0.630·2-s − 1.60·4-s + 3.89·5-s − 3.68·7-s − 2.27·8-s + 2.45·10-s + 4.96·11-s − 1.69·13-s − 2.32·14-s + 1.77·16-s − 5.52·17-s + 4.21·19-s − 6.24·20-s + 3.13·22-s + 2.77·23-s + 10.1·25-s − 1.06·26-s + 5.90·28-s + 2.31·29-s − 0.199·31-s + 5.66·32-s − 3.48·34-s − 14.3·35-s + 1.79·37-s + 2.65·38-s − 8.85·40-s + 12.1·41-s + ⋯ |
| L(s) = 1 | + 0.445·2-s − 0.801·4-s + 1.74·5-s − 1.39·7-s − 0.803·8-s + 0.777·10-s + 1.49·11-s − 0.468·13-s − 0.620·14-s + 0.442·16-s − 1.33·17-s + 0.966·19-s − 1.39·20-s + 0.667·22-s + 0.578·23-s + 2.03·25-s − 0.209·26-s + 1.11·28-s + 0.429·29-s − 0.0358·31-s + 1.00·32-s − 0.597·34-s − 2.42·35-s + 0.294·37-s + 0.431·38-s − 1.39·40-s + 1.90·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.204132368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.204132368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 0.630T + 2T^{2} \) |
| 5 | \( 1 - 3.89T + 5T^{2} \) |
| 7 | \( 1 + 3.68T + 7T^{2} \) |
| 11 | \( 1 - 4.96T + 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 + 5.52T + 17T^{2} \) |
| 19 | \( 1 - 4.21T + 19T^{2} \) |
| 23 | \( 1 - 2.77T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 + 0.199T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 + 5.34T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 1.32T + 53T^{2} \) |
| 59 | \( 1 + 5.78T + 59T^{2} \) |
| 61 | \( 1 + 0.0766T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 5.20T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 7.82T + 79T^{2} \) |
| 83 | \( 1 + 2.55T + 83T^{2} \) |
| 89 | \( 1 - 4.35T + 89T^{2} \) |
| 97 | \( 1 - 9.02T + 97T^{2} \) |
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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
−9.179678985276129169805906357332, −8.850228302807517163593374155659, −7.19730313509787437844807403366, −6.35842809573470315694051390426, −6.07444175361072807422979078799, −5.12969875932958042995824432580, −4.24860611356543718345711548355, −3.25214284843653085456393215233, −2.37196706815865171227513388678, −0.940626712707947744438495944189,
0.940626712707947744438495944189, 2.37196706815865171227513388678, 3.25214284843653085456393215233, 4.24860611356543718345711548355, 5.12969875932958042995824432580, 6.07444175361072807422979078799, 6.35842809573470315694051390426, 7.19730313509787437844807403366, 8.850228302807517163593374155659, 9.179678985276129169805906357332
