| L(s) = 1 | − 11.1·3-s + 122.·7-s − 119.·9-s + 100·11-s − 734.·13-s + 979.·17-s + 2.24e3·19-s − 1.36e3·21-s − 3.41e3·23-s + 4.03e3·27-s − 7.85e3·29-s + 2.14e3·31-s − 1.11e3·33-s + 1.04e4·37-s + 8.18e3·39-s − 7.41e3·41-s + 1.77e4·43-s + 9.43e3·47-s − 1.80e3·49-s − 1.09e4·51-s + 2.42e4·53-s − 2.49e4·57-s − 2.59e4·59-s − 3.05e3·61-s − 1.45e4·63-s − 5.87e4·67-s + 3.80e4·69-s + ⋯ |
| L(s) = 1 | − 0.714·3-s + 0.944·7-s − 0.489·9-s + 0.249·11-s − 1.20·13-s + 0.822·17-s + 1.42·19-s − 0.674·21-s − 1.34·23-s + 1.06·27-s − 1.73·29-s + 0.400·31-s − 0.178·33-s + 1.24·37-s + 0.861·39-s − 0.688·41-s + 1.46·43-s + 0.622·47-s − 0.107·49-s − 0.587·51-s + 1.18·53-s − 1.01·57-s − 0.971·59-s − 0.105·61-s − 0.462·63-s − 1.59·67-s + 0.962·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 11.1T + 243T^{2} \) |
| 7 | \( 1 - 122.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 100T + 1.61e5T^{2} \) |
| 13 | \( 1 + 734.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 979.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.24e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.04e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.77e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.43e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.42e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.05e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.76e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.62e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 826T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.75e4T + 8.58e9T^{2} \) |
| 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
−10.06476433693374402781690937415, −9.211491275119775273157434754493, −7.922428708031892592764370896946, −7.35675809772460653499090833222, −5.87206133025554809034082271379, −5.32419860775500449794077016002, −4.22229360111555653389876517937, −2.72629607185546027503657036339, −1.33477019010967319602636011112, 0,
1.33477019010967319602636011112, 2.72629607185546027503657036339, 4.22229360111555653389876517937, 5.32419860775500449794077016002, 5.87206133025554809034082271379, 7.35675809772460653499090833222, 7.922428708031892592764370896946, 9.211491275119775273157434754493, 10.06476433693374402781690937415
