| L(s) = 1 | − 0.454·2-s + 2.93·3-s − 1.79·4-s + 5-s − 1.33·6-s + 4.01·7-s + 1.72·8-s + 5.59·9-s − 0.454·10-s − 11-s − 5.25·12-s − 2.54·13-s − 1.82·14-s + 2.93·15-s + 2.80·16-s − 3.98·17-s − 2.54·18-s − 5.22·19-s − 1.79·20-s + 11.7·21-s + 0.454·22-s + 0.343·23-s + 5.05·24-s + 25-s + 1.15·26-s + 7.61·27-s − 7.19·28-s + ⋯ |
| L(s) = 1 | − 0.321·2-s + 1.69·3-s − 0.896·4-s + 0.447·5-s − 0.544·6-s + 1.51·7-s + 0.609·8-s + 1.86·9-s − 0.143·10-s − 0.301·11-s − 1.51·12-s − 0.706·13-s − 0.487·14-s + 0.757·15-s + 0.700·16-s − 0.966·17-s − 0.599·18-s − 1.19·19-s − 0.401·20-s + 2.56·21-s + 0.0968·22-s + 0.0715·23-s + 1.03·24-s + 0.200·25-s + 0.227·26-s + 1.46·27-s − 1.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.189043856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.189043856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 0.454T + 2T^{2} \) |
| 3 | \( 1 - 2.93T + 3T^{2} \) |
| 7 | \( 1 - 4.01T + 7T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 + 3.98T + 17T^{2} \) |
| 19 | \( 1 + 5.22T + 19T^{2} \) |
| 23 | \( 1 - 0.343T + 23T^{2} \) |
| 29 | \( 1 - 6.48T + 29T^{2} \) |
| 31 | \( 1 - 7.89T + 31T^{2} \) |
| 37 | \( 1 - 8.95T + 37T^{2} \) |
| 41 | \( 1 - 4.00T + 41T^{2} \) |
| 43 | \( 1 + 3.95T + 43T^{2} \) |
| 47 | \( 1 - 4.50T + 47T^{2} \) |
| 53 | \( 1 - 0.587T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 9.08T + 61T^{2} \) |
| 67 | \( 1 + 3.80T + 67T^{2} \) |
| 71 | \( 1 + 0.695T + 71T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 9.19T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 7.88T + 97T^{2} \) |
| 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
−8.538625017538391690858291406588, −8.023280072905936682150202471896, −7.43128316232497934490064531355, −6.40666681074195811327085246494, −5.08353053099650488544237438378, −4.53001756433252291613137338058, −4.01593934715421952121297520177, −2.60362699950604863274978035486, −2.18478339166499572682585970015, −1.06357970762946565629894496332,
1.06357970762946565629894496332, 2.18478339166499572682585970015, 2.60362699950604863274978035486, 4.01593934715421952121297520177, 4.53001756433252291613137338058, 5.08353053099650488544237438378, 6.40666681074195811327085246494, 7.43128316232497934490064531355, 8.023280072905936682150202471896, 8.538625017538391690858291406588
