| L(s) = 1 | − 2·2-s − 6.15·3-s + 4·4-s + 18.3·5-s + 12.3·6-s + 21.8·7-s − 8·8-s + 10.8·9-s − 36.6·10-s − 8.30·11-s − 24.6·12-s + 53.0·13-s − 43.6·14-s − 112.·15-s + 16·16-s + 74.2·17-s − 21.6·18-s − 19·19-s + 73.2·20-s − 134.·21-s + 16.6·22-s − 163.·23-s + 49.2·24-s + 210.·25-s − 106.·26-s + 99.3·27-s + 87.3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.18·3-s + 0.5·4-s + 1.63·5-s + 0.837·6-s + 1.17·7-s − 0.353·8-s + 0.401·9-s − 1.15·10-s − 0.227·11-s − 0.591·12-s + 1.13·13-s − 0.834·14-s − 1.93·15-s + 0.250·16-s + 1.05·17-s − 0.284·18-s − 0.229·19-s + 0.818·20-s − 1.39·21-s + 0.160·22-s − 1.48·23-s + 0.418·24-s + 1.68·25-s − 0.800·26-s + 0.708·27-s + 0.589·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9144572009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9144572009\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 + 6.15T + 27T^{2} \) |
| 5 | \( 1 - 18.3T + 125T^{2} \) |
| 7 | \( 1 - 21.8T + 343T^{2} \) |
| 11 | \( 1 + 8.30T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.2T + 4.91e3T^{2} \) |
| 23 | \( 1 + 163.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 232.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 98.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 296.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 171.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 366.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 138.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 572.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 632.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 183.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 56.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 68.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 332.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 368.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 426.T + 9.12e5T^{2} \) |
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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
−16.42362552439556254127211450200, −14.65462673527854666915208766640, −13.42805270694816707438894980498, −11.81518792602670282986256710435, −10.80785125592809109236580824559, −9.841794763631763200053699624608, −8.232195932100640022413740604025, −6.24615267287495912324842548824, −5.39204263415166716592434760149, −1.58422284959852460289428135042,
1.58422284959852460289428135042, 5.39204263415166716592434760149, 6.24615267287495912324842548824, 8.232195932100640022413740604025, 9.841794763631763200053699624608, 10.80785125592809109236580824559, 11.81518792602670282986256710435, 13.42805270694816707438894980498, 14.65462673527854666915208766640, 16.42362552439556254127211450200
