| L(s) = 1 | − 210.·3-s + 9.90e3·7-s + 2.44e4·9-s − 3.64e4·11-s − 1.64e5·13-s − 8.23e4·17-s + 6.09e5·19-s − 2.08e6·21-s + 1.88e6·23-s − 1.01e6·27-s + 3.39e5·29-s − 5.47e5·31-s + 7.66e6·33-s − 5.25e6·37-s + 3.46e7·39-s + 2.05e6·41-s − 6.76e6·43-s + 3.15e7·47-s + 5.77e7·49-s + 1.73e7·51-s + 4.89e7·53-s − 1.28e8·57-s − 8.77e7·59-s + 3.84e7·61-s + 2.42e8·63-s − 1.36e8·67-s − 3.96e8·69-s + ⋯ |
| L(s) = 1 | − 1.49·3-s + 1.55·7-s + 1.24·9-s − 0.750·11-s − 1.60·13-s − 0.239·17-s + 1.07·19-s − 2.33·21-s + 1.40·23-s − 0.365·27-s + 0.0890·29-s − 0.106·31-s + 1.12·33-s − 0.461·37-s + 2.39·39-s + 0.113·41-s − 0.301·43-s + 0.942·47-s + 1.43·49-s + 0.358·51-s + 0.852·53-s − 1.60·57-s − 0.943·59-s + 0.355·61-s + 1.94·63-s − 0.825·67-s − 2.10·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.165415521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.165415521\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 210.T + 1.96e4T^{2} \) |
| 7 | \( 1 - 9.90e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.64e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.64e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 8.23e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.09e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.88e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.39e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.47e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 5.25e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.05e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 6.76e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.15e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.89e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.77e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.84e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.36e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.49e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.61e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.26e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.87e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.63e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.71e8T + 7.60e17T^{2} \) |
| 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
−10.11177440682258254095848211940, −8.859527198975596252719615919867, −7.58654151190214447615255615600, −7.12301446298561943025059427563, −5.67171110586883257374928222831, −5.04538080198705077971377431449, −4.58113061728645605786661401964, −2.72072035432709384259691462090, −1.49110649708044709369217794746, −0.51251298821951538131137769755,
0.51251298821951538131137769755, 1.49110649708044709369217794746, 2.72072035432709384259691462090, 4.58113061728645605786661401964, 5.04538080198705077971377431449, 5.67171110586883257374928222831, 7.12301446298561943025059427563, 7.58654151190214447615255615600, 8.859527198975596252719615919867, 10.11177440682258254095848211940
