L(s) = 1 | + 10.8·2-s + 85.3·4-s − 25·5-s − 188.·7-s + 578.·8-s − 270.·10-s − 121·11-s + 1.06e3·13-s − 2.04e3·14-s + 3.53e3·16-s + 2.03e3·17-s + 285.·19-s − 2.13e3·20-s − 1.31e3·22-s + 589.·23-s + 625·25-s + 1.15e4·26-s − 1.60e4·28-s − 5.68e3·29-s + 1.89e3·31-s + 1.97e4·32-s + 2.20e4·34-s + 4.71e3·35-s + 6.51e3·37-s + 3.09e3·38-s − 1.44e4·40-s + 1.75e4·41-s + ⋯ |
L(s) = 1 | + 1.91·2-s + 2.66·4-s − 0.447·5-s − 1.45·7-s + 3.19·8-s − 0.856·10-s − 0.301·11-s + 1.74·13-s − 2.78·14-s + 3.44·16-s + 1.70·17-s + 0.181·19-s − 1.19·20-s − 0.577·22-s + 0.232·23-s + 0.200·25-s + 3.33·26-s − 3.87·28-s − 1.25·29-s + 0.354·31-s + 3.41·32-s + 3.26·34-s + 0.650·35-s + 0.782·37-s + 0.347·38-s − 1.42·40-s + 1.63·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(7.303687049\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.303687049\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
good | 2 | \( 1 - 10.8T + 32T^{2} \) |
| 7 | \( 1 + 188.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.06e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 285.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 589.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.68e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.89e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.51e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.75e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.94e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.80e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 360.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.27e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.22e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.78e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.80e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.15e5T + 8.58e9T^{2} \) |
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
−10.59466250680175392044525128744, −9.452404586584157570322363880541, −7.915083596748896428551819941580, −7.06514048324255767127799256643, −5.99526423227046160571268553955, −5.67527881429888931593046640205, −4.14449872562013256903446968734, −3.49930818419237321492696597111, −2.78940112843506281793410312400, −1.08756825079484451260783491760,
1.08756825079484451260783491760, 2.78940112843506281793410312400, 3.49930818419237321492696597111, 4.14449872562013256903446968734, 5.67527881429888931593046640205, 5.99526423227046160571268553955, 7.06514048324255767127799256643, 7.915083596748896428551819941580, 9.452404586584157570322363880541, 10.59466250680175392044525128744