| L(s) = 1 | − 10.7·2-s − 23.6·3-s + 83.9·4-s + 70.6·5-s + 254.·6-s − 64.6·7-s − 558.·8-s + 317.·9-s − 760.·10-s − 47.9·11-s − 1.98e3·12-s + 1.00e3·13-s + 695.·14-s − 1.67e3·15-s + 3.33e3·16-s − 1.77e3·17-s − 3.41e3·18-s − 910.·19-s + 5.92e3·20-s + 1.52e3·21-s + 516.·22-s − 4.17e3·23-s + 1.32e4·24-s + 1.86e3·25-s − 1.07e4·26-s − 1.75e3·27-s − 5.42e3·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 1.51·3-s + 2.62·4-s + 1.26·5-s + 2.88·6-s − 0.498·7-s − 3.08·8-s + 1.30·9-s − 2.40·10-s − 0.119·11-s − 3.98·12-s + 1.64·13-s + 0.948·14-s − 1.91·15-s + 3.25·16-s − 1.49·17-s − 2.48·18-s − 0.578·19-s + 3.31·20-s + 0.756·21-s + 0.227·22-s − 1.64·23-s + 4.68·24-s + 0.597·25-s − 3.12·26-s − 0.463·27-s − 1.30·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 1.36e3T \) |
| good | 2 | \( 1 + 10.7T + 32T^{2} \) |
| 3 | \( 1 + 23.6T + 243T^{2} \) |
| 5 | \( 1 - 70.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 64.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 47.9T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.00e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.77e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 910.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.17e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.61e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.85e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + 7.98e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.02e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.64e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.31e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.03e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.99e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 278.T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.72e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.30e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.73e4T + 8.58e9T^{2} \) |
| 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
−15.70969553958092880894578239308, −13.24099095087065526196431765585, −11.64801091989522385434585290865, −10.69923284733650593816299973248, −9.924537201210624973342448724054, −8.619391096155637491283398620480, −6.51112998817414404519069734727, −6.07644226310745780541732569965, −1.72269788420501144990646336257, 0,
1.72269788420501144990646336257, 6.07644226310745780541732569965, 6.51112998817414404519069734727, 8.619391096155637491283398620480, 9.924537201210624973342448724054, 10.69923284733650593816299973248, 11.64801091989522385434585290865, 13.24099095087065526196431765585, 15.70969553958092880894578239308
