| L(s) = 1 | + 0.438·2-s − 31.8·4-s − 61.4·5-s − 162.·7-s − 27.9·8-s − 26.9·10-s + 361.·11-s − 169·13-s − 71.1·14-s + 1.00e3·16-s + 1.57e3·17-s − 98.2·19-s + 1.95e3·20-s + 158.·22-s − 1.60e3·23-s + 652.·25-s − 74.0·26-s + 5.16e3·28-s + 307.·29-s + 2.93e3·31-s + 1.33e3·32-s + 692.·34-s + 9.97e3·35-s − 1.22e4·37-s − 43.0·38-s + 1.71e3·40-s + 104.·41-s + ⋯ |
| L(s) = 1 | + 0.0775·2-s − 0.993·4-s − 1.09·5-s − 1.25·7-s − 0.154·8-s − 0.0852·10-s + 0.899·11-s − 0.277·13-s − 0.0970·14-s + 0.982·16-s + 1.32·17-s − 0.0624·19-s + 1.09·20-s + 0.0697·22-s − 0.633·23-s + 0.208·25-s − 0.0214·26-s + 1.24·28-s + 0.0677·29-s + 0.548·31-s + 0.230·32-s + 0.102·34-s + 1.37·35-s − 1.46·37-s − 0.00484·38-s + 0.169·40-s + 0.00967·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8199310818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8199310818\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + 169T \) |
| good | 2 | \( 1 - 0.438T + 32T^{2} \) |
| 5 | \( 1 + 61.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 162.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 361.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.57e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 98.2T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 307.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.93e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.22e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 104.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.49e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.59e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.59e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.01e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.53e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.61e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.54e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.57e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 912.T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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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
−12.39063869080826074976230821740, −11.98701821259515519252856129782, −10.27342178450181489854561651443, −9.410347533601409187427558189270, −8.343360182806907972151216493741, −7.15951526330041339330564534067, −5.73991203473890242443493145600, −4.14740036752820412454032315796, −3.37522108715715801591009175642, −0.61920917800492875048664976982,
0.61920917800492875048664976982, 3.37522108715715801591009175642, 4.14740036752820412454032315796, 5.73991203473890242443493145600, 7.15951526330041339330564534067, 8.343360182806907972151216493741, 9.410347533601409187427558189270, 10.27342178450181489854561651443, 11.98701821259515519252856129782, 12.39063869080826074976230821740
