| L(s) = 1 | + 10.2·2-s + 72.3·4-s + 81.8·5-s − 15.0·7-s + 412.·8-s + 836.·10-s − 292.·11-s + 859.·13-s − 153.·14-s + 1.89e3·16-s − 1.05e3·17-s + 1.55e3·19-s + 5.92e3·20-s − 2.99e3·22-s + 2.86e3·23-s + 3.57e3·25-s + 8.77e3·26-s − 1.08e3·28-s − 8.56e3·29-s + 6.39e3·31-s + 6.16e3·32-s − 1.07e4·34-s − 1.23e3·35-s + 348.·37-s + 1.58e4·38-s + 3.37e4·40-s − 1.68e3·41-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 2.26·4-s + 1.46·5-s − 0.116·7-s + 2.27·8-s + 2.64·10-s − 0.729·11-s + 1.41·13-s − 0.209·14-s + 1.85·16-s − 0.881·17-s + 0.985·19-s + 3.31·20-s − 1.31·22-s + 1.13·23-s + 1.14·25-s + 2.54·26-s − 0.262·28-s − 1.89·29-s + 1.19·31-s + 1.06·32-s − 1.59·34-s − 0.169·35-s + 0.0419·37-s + 1.77·38-s + 3.33·40-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(8.967479705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.967479705\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 41 | \( 1 + 1.68e3T \) |
| good | 2 | \( 1 - 10.2T + 32T^{2} \) |
| 5 | \( 1 - 81.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 15.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 292.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 859.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.05e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.55e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.86e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.56e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.39e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 348.T + 6.93e7T^{2} \) |
| 43 | \( 1 + 1.23e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.54e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.65e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.14e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.50e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.31e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.43e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.60e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.92022378051293105355647196244, −9.929344568564107602533816872389, −8.773400901872357944244030792880, −7.22540583892436736648007360733, −6.27670490897547442923569734939, −5.64058360393861375527346812663, −4.85649694909172797921356946367, −3.51822524114226492440737598677, −2.54082407112567562883298326723, −1.47719898669922113957199324354,
1.47719898669922113957199324354, 2.54082407112567562883298326723, 3.51822524114226492440737598677, 4.85649694909172797921356946367, 5.64058360393861375527346812663, 6.27670490897547442923569734939, 7.22540583892436736648007360733, 8.773400901872357944244030792880, 9.929344568564107602533816872389, 10.92022378051293105355647196244
