| L(s) = 1 | − 2-s + 2.80·3-s + 4-s + 4.09·5-s − 2.80·6-s + 7-s − 8-s + 4.89·9-s − 4.09·10-s + 5.04·11-s + 2.80·12-s − 6.06·13-s − 14-s + 11.4·15-s + 16-s − 1.24·17-s − 4.89·18-s − 3.98·19-s + 4.09·20-s + 2.80·21-s − 5.04·22-s + 7.27·23-s − 2.80·24-s + 11.7·25-s + 6.06·26-s + 5.31·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.62·3-s + 0.5·4-s + 1.82·5-s − 1.14·6-s + 0.377·7-s − 0.353·8-s + 1.63·9-s − 1.29·10-s + 1.52·11-s + 0.810·12-s − 1.68·13-s − 0.267·14-s + 2.96·15-s + 0.250·16-s − 0.301·17-s − 1.15·18-s − 0.913·19-s + 0.914·20-s + 0.613·21-s − 1.07·22-s + 1.51·23-s − 0.573·24-s + 2.34·25-s + 1.18·26-s + 1.02·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.323486468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.323486468\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 2.80T + 3T^{2} \) |
| 5 | \( 1 - 4.09T + 5T^{2} \) |
| 11 | \( 1 - 5.04T + 11T^{2} \) |
| 13 | \( 1 + 6.06T + 13T^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 19 | \( 1 + 3.98T + 19T^{2} \) |
| 23 | \( 1 - 7.27T + 23T^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 31 | \( 1 - 7.06T + 31T^{2} \) |
| 37 | \( 1 + 0.682T + 37T^{2} \) |
| 41 | \( 1 - 1.11T + 41T^{2} \) |
| 43 | \( 1 + 8.81T + 43T^{2} \) |
| 47 | \( 1 + 7.64T + 47T^{2} \) |
| 53 | \( 1 - 0.364T + 53T^{2} \) |
| 59 | \( 1 + 7.49T + 59T^{2} \) |
| 61 | \( 1 + 0.389T + 61T^{2} \) |
| 67 | \( 1 - 7.81T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 2.73T + 73T^{2} \) |
| 79 | \( 1 - 17.7T + 79T^{2} \) |
| 83 | \( 1 + 8.83T + 83T^{2} \) |
| 89 | \( 1 + 3.96T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−8.369871050718652394310445823907, −7.45309648165159030125332651992, −6.70271887832293841215835664880, −6.37944042734151245608969165043, −5.08941678929480415790237867081, −4.47638436521966927015853366090, −3.18487430053389933356152269396, −2.53964620359543884777080671685, −1.91890370126133668582105655141, −1.26565045235882911831461786121,
1.26565045235882911831461786121, 1.91890370126133668582105655141, 2.53964620359543884777080671685, 3.18487430053389933356152269396, 4.47638436521966927015853366090, 5.08941678929480415790237867081, 6.37944042734151245608969165043, 6.70271887832293841215835664880, 7.45309648165159030125332651992, 8.369871050718652394310445823907
