| L(s) = 1 | − 2.27·2-s + 3.20·3-s + 3.19·4-s + 0.137·5-s − 7.31·6-s + 1.86·7-s − 2.72·8-s + 7.30·9-s − 0.313·10-s + 2.62·11-s + 10.2·12-s − 1.74·13-s − 4.26·14-s + 0.441·15-s − 0.171·16-s + 1.52·17-s − 16.6·18-s + 4.89·19-s + 0.440·20-s + 6.00·21-s − 5.98·22-s + 4.37·23-s − 8.76·24-s − 4.98·25-s + 3.97·26-s + 13.8·27-s + 5.97·28-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.85·3-s + 1.59·4-s + 0.0615·5-s − 2.98·6-s + 0.706·7-s − 0.965·8-s + 2.43·9-s − 0.0992·10-s + 0.791·11-s + 2.96·12-s − 0.484·13-s − 1.13·14-s + 0.114·15-s − 0.0429·16-s + 0.369·17-s − 3.92·18-s + 1.12·19-s + 0.0984·20-s + 1.30·21-s − 1.27·22-s + 0.911·23-s − 1.78·24-s − 0.996·25-s + 0.780·26-s + 2.65·27-s + 1.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.927595706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.927595706\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 3 | \( 1 - 3.20T + 3T^{2} \) |
| 5 | \( 1 - 0.137T + 5T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 - 2.62T + 11T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 - 1.52T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 - 0.0316T + 29T^{2} \) |
| 31 | \( 1 + 2.83T + 31T^{2} \) |
| 37 | \( 1 + 0.00930T + 37T^{2} \) |
| 41 | \( 1 + 8.32T + 41T^{2} \) |
| 47 | \( 1 + 5.62T + 47T^{2} \) |
| 53 | \( 1 - 7.73T + 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 7.73T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 4.75T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 4.30T + 83T^{2} \) |
| 89 | \( 1 + 7.69T + 89T^{2} \) |
| 97 | \( 1 - 1.61T + 97T^{2} \) |
| 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
−9.105226081428472335441243374863, −8.582747552464541441123941558077, −7.893979459432297709201967579798, −7.38784845179111224729062564546, −6.73221178797126508342334503986, −5.09671476966260000936533649141, −3.92227768507244974597919099170, −2.96595396303834600362790474524, −1.94412552565439090898913543399, −1.25627898504113589823191747540,
1.25627898504113589823191747540, 1.94412552565439090898913543399, 2.96595396303834600362790474524, 3.92227768507244974597919099170, 5.09671476966260000936533649141, 6.73221178797126508342334503986, 7.38784845179111224729062564546, 7.893979459432297709201967579798, 8.582747552464541441123941558077, 9.105226081428472335441243374863
