| L(s) = 1 | + 1.85·2-s + 1.64·3-s + 1.42·4-s + 3.93·5-s + 3.04·6-s − 3.53·7-s − 1.05·8-s − 0.291·9-s + 7.28·10-s + 2.86·11-s + 2.35·12-s − 5.08·13-s − 6.54·14-s + 6.47·15-s − 4.81·16-s − 2.89·17-s − 0.540·18-s + 1.67·19-s + 5.61·20-s − 5.81·21-s + 5.31·22-s + 0.133·23-s − 1.73·24-s + 10.4·25-s − 9.40·26-s − 5.41·27-s − 5.04·28-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.950·3-s + 0.714·4-s + 1.75·5-s + 1.24·6-s − 1.33·7-s − 0.373·8-s − 0.0972·9-s + 2.30·10-s + 0.864·11-s + 0.678·12-s − 1.40·13-s − 1.74·14-s + 1.67·15-s − 1.20·16-s − 0.702·17-s − 0.127·18-s + 0.384·19-s + 1.25·20-s − 1.26·21-s + 1.13·22-s + 0.0279·23-s − 0.355·24-s + 2.09·25-s − 1.84·26-s − 1.04·27-s − 0.954·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.517501505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.517501505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 409 | \( 1 - T \) |
| good | 2 | \( 1 - 1.85T + 2T^{2} \) |
| 3 | \( 1 - 1.64T + 3T^{2} \) |
| 5 | \( 1 - 3.93T + 5T^{2} \) |
| 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 13 | \( 1 + 5.08T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 - 1.67T + 19T^{2} \) |
| 23 | \( 1 - 0.133T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 31 | \( 1 - 2.50T + 31T^{2} \) |
| 37 | \( 1 - 9.64T + 37T^{2} \) |
| 41 | \( 1 + 0.232T + 41T^{2} \) |
| 43 | \( 1 - 5.40T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 8.53T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 9.21T + 67T^{2} \) |
| 71 | \( 1 - 8.87T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 6.15T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 - 8.70T + 97T^{2} \) |
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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
−11.53542041302173120368463078158, −9.966523214857130557318155241943, −9.457280453799635317828961094589, −8.903337176349926263232918892328, −7.07640166779941968861276684676, −6.22711497174664573266598226157, −5.54472290771592560946165740539, −4.24299479208125124719959549486, −2.92574732751907403665649154467, −2.38743188562189126234024893368,
2.38743188562189126234024893368, 2.92574732751907403665649154467, 4.24299479208125124719959549486, 5.54472290771592560946165740539, 6.22711497174664573266598226157, 7.07640166779941968861276684676, 8.903337176349926263232918892328, 9.457280453799635317828961094589, 9.966523214857130557318155241943, 11.53542041302173120368463078158
