| L(s) = 1 | − 3.62·2-s + 9·3-s − 18.8·4-s − 32.5·6-s − 168.·7-s + 184.·8-s + 81·9-s + 121·11-s − 169.·12-s + 155.·13-s + 608.·14-s − 63.3·16-s + 426.·17-s − 293.·18-s − 1.67e3·19-s − 1.51e3·21-s − 438.·22-s + 11.2·23-s + 1.65e3·24-s − 561.·26-s + 729·27-s + 3.17e3·28-s − 1.10e3·29-s + 7.18e3·31-s − 5.66e3·32-s + 1.08e3·33-s − 1.54e3·34-s + ⋯ |
| L(s) = 1 | − 0.640·2-s + 0.577·3-s − 0.590·4-s − 0.369·6-s − 1.29·7-s + 1.01·8-s + 0.333·9-s + 0.301·11-s − 0.340·12-s + 0.254·13-s + 0.829·14-s − 0.0618·16-s + 0.358·17-s − 0.213·18-s − 1.06·19-s − 0.748·21-s − 0.193·22-s + 0.00442·23-s + 0.587·24-s − 0.162·26-s + 0.192·27-s + 0.764·28-s − 0.244·29-s + 1.34·31-s − 0.978·32-s + 0.174·33-s − 0.229·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9735828520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9735828520\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 3.62T + 32T^{2} \) |
| 7 | \( 1 + 168.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 155.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 426.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.67e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 11.2T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.10e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.18e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.57e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.40e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.70e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 916.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.53e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.67e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.59e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.75e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.44e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.20e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.57e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.658952179350844657699515028465, −8.568857207966434233793850809976, −8.164965257290873628445211635085, −6.94699287369959529570304144021, −6.25746022053161501281865841680, −4.89759557849990516959039227663, −3.88878906870623158335138099487, −3.07677448214362992182670673341, −1.70733469467946204101653446186, −0.48558607229114015980491567568,
0.48558607229114015980491567568, 1.70733469467946204101653446186, 3.07677448214362992182670673341, 3.88878906870623158335138099487, 4.89759557849990516959039227663, 6.25746022053161501281865841680, 6.94699287369959529570304144021, 8.164965257290873628445211635085, 8.568857207966434233793850809976, 9.658952179350844657699515028465
