| L(s) = 1 | − 2.56·2-s + 0.729·3-s + 4.59·4-s − 3.21·5-s − 1.87·6-s + 3.04·7-s − 6.65·8-s − 2.46·9-s + 8.25·10-s + 3.60·11-s + 3.35·12-s + 5.51·13-s − 7.81·14-s − 2.34·15-s + 7.90·16-s − 4.15·17-s + 6.33·18-s − 19-s − 14.7·20-s + 2.22·21-s − 9.26·22-s − 1.11·23-s − 4.85·24-s + 5.34·25-s − 14.1·26-s − 3.98·27-s + 13.9·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 0.421·3-s + 2.29·4-s − 1.43·5-s − 0.764·6-s + 1.15·7-s − 2.35·8-s − 0.822·9-s + 2.61·10-s + 1.08·11-s + 0.967·12-s + 1.52·13-s − 2.08·14-s − 0.605·15-s + 1.97·16-s − 1.00·17-s + 1.49·18-s − 0.229·19-s − 3.30·20-s + 0.484·21-s − 1.97·22-s − 0.233·23-s − 0.991·24-s + 1.06·25-s − 2.77·26-s − 0.767·27-s + 2.64·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7638824808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7638824808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 317 | \( 1 - T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 - 0.729T + 3T^{2} \) |
| 5 | \( 1 + 3.21T + 5T^{2} \) |
| 7 | \( 1 - 3.04T + 7T^{2} \) |
| 11 | \( 1 - 3.60T + 11T^{2} \) |
| 13 | \( 1 - 5.51T + 13T^{2} \) |
| 17 | \( 1 + 4.15T + 17T^{2} \) |
| 23 | \( 1 + 1.11T + 23T^{2} \) |
| 29 | \( 1 + 1.00T + 29T^{2} \) |
| 31 | \( 1 - 5.28T + 31T^{2} \) |
| 37 | \( 1 - 0.858T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 8.43T + 43T^{2} \) |
| 47 | \( 1 - 5.80T + 47T^{2} \) |
| 53 | \( 1 - 8.35T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 + 7.23T + 73T^{2} \) |
| 79 | \( 1 - 7.64T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 3.63T + 89T^{2} \) |
| 97 | \( 1 - 3.45T + 97T^{2} \) |
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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
−8.141803159311612529377876691586, −7.944117484004801759394200592647, −6.79237433521264847222649311233, −6.56202062522573140549281799843, −5.32200345555904076823743582153, −4.08383611600277181965696863867, −3.60070611949526556075063683341, −2.42849463497773236436986138234, −1.53879112229474323959015653986, −0.62158263254706991589149728000,
0.62158263254706991589149728000, 1.53879112229474323959015653986, 2.42849463497773236436986138234, 3.60070611949526556075063683341, 4.08383611600277181965696863867, 5.32200345555904076823743582153, 6.56202062522573140549281799843, 6.79237433521264847222649311233, 7.944117484004801759394200592647, 8.141803159311612529377876691586
