| L(s) = 1 | + 2.61·2-s + 0.874·3-s + 4.85·4-s + 2.23·5-s + 2.28·6-s − 7-s + 7.47·8-s − 2.23·9-s + 5.85·10-s − 4.24·11-s + 4.24·12-s + 2.62·13-s − 2.61·14-s + 1.95·15-s + 9.85·16-s − 3.70·17-s − 5.85·18-s + 19-s + 10.8·20-s − 0.874·21-s − 11.1·22-s + 2.62·23-s + 6.53·24-s + 6.86·26-s − 4.57·27-s − 4.85·28-s + 0.540·29-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 0.504·3-s + 2.42·4-s + 0.999·5-s + 0.934·6-s − 0.377·7-s + 2.64·8-s − 0.745·9-s + 1.85·10-s − 1.27·11-s + 1.22·12-s + 0.727·13-s − 0.699·14-s + 0.504·15-s + 2.46·16-s − 0.897·17-s − 1.37·18-s + 0.229·19-s + 2.42·20-s − 0.190·21-s − 2.36·22-s + 0.546·23-s + 1.33·24-s + 1.34·26-s − 0.880·27-s − 0.917·28-s + 0.100·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.730760521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.730760521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 3 | \( 1 - 0.874T + 3T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 - 0.540T + 29T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 + 1.47T + 41T^{2} \) |
| 43 | \( 1 - 9.69T + 43T^{2} \) |
| 47 | \( 1 + 9.70T + 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 6T + 67T^{2} \) |
| 71 | \( 1 + 1.47T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 - 1.62T + 79T^{2} \) |
| 83 | \( 1 - 3.16T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
| 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
−10.34141534375182547560963304805, −9.220689906391795634732512557006, −8.220165803696698982468529932027, −7.13056377348582072459432912682, −6.20316807247412351608524746131, −5.62248721527892227193494269627, −4.86869594814239585919198030997, −3.61996820034014468913045259756, −2.77893269315773244338193434759, −2.06877405223618416776328319547,
2.06877405223618416776328319547, 2.77893269315773244338193434759, 3.61996820034014468913045259756, 4.86869594814239585919198030997, 5.62248721527892227193494269627, 6.20316807247412351608524746131, 7.13056377348582072459432912682, 8.220165803696698982468529932027, 9.220689906391795634732512557006, 10.34141534375182547560963304805
