| L(s) = 1 | − 4.38·2-s + 7.64·3-s + 11.1·4-s − 5·5-s − 33.4·6-s + 9.94·7-s − 13.9·8-s + 31.4·9-s + 21.9·10-s − 11·11-s + 85.5·12-s − 33.3·13-s − 43.5·14-s − 38.2·15-s − 28.3·16-s − 16.7·17-s − 137.·18-s − 19·19-s − 55.9·20-s + 76.0·21-s + 48.1·22-s + 190.·23-s − 106.·24-s + 25·25-s + 146.·26-s + 34.1·27-s + 111.·28-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.47·3-s + 1.39·4-s − 0.447·5-s − 2.27·6-s + 0.537·7-s − 0.617·8-s + 1.16·9-s + 0.692·10-s − 0.301·11-s + 2.05·12-s − 0.711·13-s − 0.831·14-s − 0.658·15-s − 0.442·16-s − 0.238·17-s − 1.80·18-s − 0.229·19-s − 0.625·20-s + 0.790·21-s + 0.466·22-s + 1.72·23-s − 0.908·24-s + 0.200·25-s + 1.10·26-s + 0.243·27-s + 0.751·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.478532827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.478532827\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 4.38T + 8T^{2} \) |
| 3 | \( 1 - 7.64T + 27T^{2} \) |
| 7 | \( 1 - 9.94T + 343T^{2} \) |
| 13 | \( 1 + 33.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.7T + 4.91e3T^{2} \) |
| 23 | \( 1 - 190.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 198.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 96.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 20.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 212.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 189.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 135.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 440.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 537.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 830.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 264.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 410.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 511.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 38.9T + 4.93e5T^{2} \) |
| 83 | \( 1 + 50.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 646.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 706.T + 9.12e5T^{2} \) |
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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
−9.326355437163254318718080504681, −8.690374526800002983745450803998, −8.096111648475112908236206951853, −7.47208926653197426999628593016, −6.86479243405211088776640296016, −5.11872049576219262691505786424, −3.98190776819178122385312471547, −2.74714631659590659560918376877, −2.02254072579745050171565671477, −0.75300420436142968478195031384,
0.75300420436142968478195031384, 2.02254072579745050171565671477, 2.74714631659590659560918376877, 3.98190776819178122385312471547, 5.11872049576219262691505786424, 6.86479243405211088776640296016, 7.47208926653197426999628593016, 8.096111648475112908236206951853, 8.690374526800002983745450803998, 9.326355437163254318718080504681
