| L(s) = 1 | − 2.30·2-s + 0.621·3-s + 3.29·4-s − 1.43·6-s + 0.210·7-s − 2.99·8-s − 2.61·9-s + 2.40·11-s + 2.05·12-s + 0.974·13-s − 0.485·14-s + 0.287·16-s − 0.931·17-s + 6.01·18-s − 4.45·19-s + 0.131·21-s − 5.53·22-s − 3.73·23-s − 1.85·24-s − 2.24·26-s − 3.48·27-s + 0.695·28-s − 6.80·29-s − 10.0·31-s + 5.32·32-s + 1.49·33-s + 2.14·34-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 0.358·3-s + 1.64·4-s − 0.584·6-s + 0.0797·7-s − 1.05·8-s − 0.871·9-s + 0.724·11-s + 0.591·12-s + 0.270·13-s − 0.129·14-s + 0.0718·16-s − 0.225·17-s + 1.41·18-s − 1.02·19-s + 0.0286·21-s − 1.17·22-s − 0.777·23-s − 0.379·24-s − 0.440·26-s − 0.671·27-s + 0.131·28-s − 1.26·29-s − 1.80·31-s + 0.940·32-s + 0.259·33-s + 0.367·34-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
| \(L(1)\) |
\(\approx\) |
\(0.6449728558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6449728558\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 - 0.621T + 3T^{2} \) |
| 7 | \( 1 - 0.210T + 7T^{2} \) |
| 11 | \( 1 - 2.40T + 11T^{2} \) |
| 13 | \( 1 - 0.974T + 13T^{2} \) |
| 17 | \( 1 + 0.931T + 17T^{2} \) |
| 19 | \( 1 + 4.45T + 19T^{2} \) |
| 23 | \( 1 + 3.73T + 23T^{2} \) |
| 29 | \( 1 + 6.80T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 6.37T + 37T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 43 | \( 1 - 4.46T + 43T^{2} \) |
| 47 | \( 1 - 3.60T + 47T^{2} \) |
| 53 | \( 1 + 9.36T + 53T^{2} \) |
| 59 | \( 1 - 0.136T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 + 1.71T + 67T^{2} \) |
| 71 | \( 1 - 8.78T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 2.51T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 0.0938T + 89T^{2} \) |
| 97 | \( 1 + 5.72T + 97T^{2} \) |
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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
−8.103305347508832983265636251438, −7.77048234140254354898254884144, −6.85404168727511123630999143480, −6.23446613389239007114424596912, −5.48955072430084542899354483871, −4.22908277674534058139577996240, −3.48726388565189176244405405209, −2.29458420291091566076458677707, −1.81089903510902427415393257341, −0.51556559028140624242584310852,
0.51556559028140624242584310852, 1.81089903510902427415393257341, 2.29458420291091566076458677707, 3.48726388565189176244405405209, 4.22908277674534058139577996240, 5.48955072430084542899354483871, 6.23446613389239007114424596912, 6.85404168727511123630999143480, 7.77048234140254354898254884144, 8.103305347508832983265636251438
