L(s) = 1 | + 10.2·2-s − 9·3-s + 73.4·4-s − 92.4·6-s + 129.·7-s + 426.·8-s + 81·9-s − 121·11-s − 661.·12-s − 26.3·13-s + 1.33e3·14-s + 2.02e3·16-s − 1.41e3·17-s + 831.·18-s + 2.33e3·19-s − 1.16e3·21-s − 1.24e3·22-s + 939.·23-s − 3.83e3·24-s − 270.·26-s − 729·27-s + 9.54e3·28-s + 6.49e3·29-s + 3.50e3·31-s + 7.15e3·32-s + 1.08e3·33-s − 1.45e4·34-s + ⋯ |
L(s) = 1 | + 1.81·2-s − 0.577·3-s + 2.29·4-s − 1.04·6-s + 1.00·7-s + 2.35·8-s + 0.333·9-s − 0.301·11-s − 1.32·12-s − 0.0432·13-s + 1.81·14-s + 1.97·16-s − 1.18·17-s + 0.605·18-s + 1.48·19-s − 0.578·21-s − 0.547·22-s + 0.370·23-s − 1.35·24-s − 0.0785·26-s − 0.192·27-s + 2.30·28-s + 1.43·29-s + 0.654·31-s + 1.23·32-s + 0.174·33-s − 2.15·34-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
\(L(3)\) |
\(\approx\) |
\(7.561118408\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.561118408\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 - 10.2T + 32T^{2} \) |
| 7 | \( 1 - 129.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 26.3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.41e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.33e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 939.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.49e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.50e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.55e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.21e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.06e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.10e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.75e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.23e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.05e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.72e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.05e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.57e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.48e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.85e4T + 8.58e9T^{2} \) |
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
−9.776948901534767507324747696960, −8.324986712714912520840880112639, −7.37798349475604195042789810527, −6.56477418524139285702583932730, −5.72986886731827354629896252464, −4.80046573150741480781362786155, −4.53144872819524230389171808441, −3.19864886352344614485801750953, −2.22084485196809432002511854716, −1.02601573771067402209216837316,
1.02601573771067402209216837316, 2.22084485196809432002511854716, 3.19864886352344614485801750953, 4.53144872819524230389171808441, 4.80046573150741480781362786155, 5.72986886731827354629896252464, 6.56477418524139285702583932730, 7.37798349475604195042789810527, 8.324986712714912520840880112639, 9.776948901534767507324747696960