| L(s) = 1 | − 2.40·2-s − 4.57·3-s − 2.20·4-s + 18.4·5-s + 11.0·6-s + 11.7·7-s + 24.5·8-s − 6.05·9-s − 44.4·10-s − 11·11-s + 10.0·12-s − 28.2·14-s − 84.5·15-s − 41.5·16-s + 129.·17-s + 14.5·18-s + 45.1·19-s − 40.6·20-s − 53.6·21-s + 26.4·22-s − 95.0·23-s − 112.·24-s + 216.·25-s + 151.·27-s − 25.8·28-s − 10.2·29-s + 203.·30-s + ⋯ |
| L(s) = 1 | − 0.851·2-s − 0.880·3-s − 0.275·4-s + 1.65·5-s + 0.749·6-s + 0.633·7-s + 1.08·8-s − 0.224·9-s − 1.40·10-s − 0.301·11-s + 0.242·12-s − 0.539·14-s − 1.45·15-s − 0.648·16-s + 1.84·17-s + 0.190·18-s + 0.545·19-s − 0.454·20-s − 0.557·21-s + 0.256·22-s − 0.861·23-s − 0.956·24-s + 1.72·25-s + 1.07·27-s − 0.174·28-s − 0.0659·29-s + 1.23·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.40T + 8T^{2} \) |
| 3 | \( 1 + 4.57T + 27T^{2} \) |
| 5 | \( 1 - 18.4T + 125T^{2} \) |
| 7 | \( 1 - 11.7T + 343T^{2} \) |
| 17 | \( 1 - 129.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 45.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 95.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 10.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 317.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 71.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 144.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 252.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 44.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 552.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 432.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 432.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 433.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 522.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 752.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 707.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 253.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.62e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.54e3T + 9.12e5T^{2} \) |
| 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
−8.552538469135593132082711602325, −7.87149384556420147527592549994, −6.94380344840962115824587007623, −5.78635822069740645080477203208, −5.47874403136172428747971524974, −4.79706650468367472053397289551, −3.28125753724261434822234470739, −1.84740195539534174336531356957, −1.24326037211165534279921125397, 0,
1.24326037211165534279921125397, 1.84740195539534174336531356957, 3.28125753724261434822234470739, 4.79706650468367472053397289551, 5.47874403136172428747971524974, 5.78635822069740645080477203208, 6.94380344840962115824587007623, 7.87149384556420147527592549994, 8.552538469135593132082711602325
