| L(s) = 1 | + 9·3-s + 82.3·5-s − 101.·7-s + 81·9-s − 154.·11-s + 169·13-s + 741.·15-s − 2.21e3·17-s − 171.·19-s − 910.·21-s + 2.49e3·23-s + 3.65e3·25-s + 729·27-s + 5.53e3·29-s + 7.17e3·31-s − 1.39e3·33-s − 8.32e3·35-s − 580.·37-s + 1.52e3·39-s + 1.18e4·41-s + 6.59e3·43-s + 6.67e3·45-s − 6.91e3·47-s − 6.58e3·49-s − 1.99e4·51-s + 3.51e4·53-s − 1.27e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.47·5-s − 0.779·7-s + 0.333·9-s − 0.385·11-s + 0.277·13-s + 0.850·15-s − 1.85·17-s − 0.108·19-s − 0.450·21-s + 0.982·23-s + 1.16·25-s + 0.192·27-s + 1.22·29-s + 1.34·31-s − 0.222·33-s − 1.14·35-s − 0.0697·37-s + 0.160·39-s + 1.10·41-s + 0.544·43-s + 0.491·45-s − 0.456·47-s − 0.391·49-s − 1.07·51-s + 1.72·53-s − 0.567·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.403576544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.403576544\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 13 | \( 1 - 169T \) |
| good | 5 | \( 1 - 82.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 101.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 154.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 2.21e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 171.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.49e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.53e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.17e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 580.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.18e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.59e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.91e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.51e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.14e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.90e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.95e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.66e4T + 8.58e9T^{2} \) |
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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
−9.805353329081282244670300954756, −9.021408372978754718610141754146, −8.378764887855950169310935047772, −6.86128270965077973696608550181, −6.43369494193725334487730861939, −5.33578369533347663109905291754, −4.23681725318260795718280439817, −2.79082975054415023070699072984, −2.25417232854210174975124078679, −0.864611366487280071582463196077,
0.864611366487280071582463196077, 2.25417232854210174975124078679, 2.79082975054415023070699072984, 4.23681725318260795718280439817, 5.33578369533347663109905291754, 6.43369494193725334487730861939, 6.86128270965077973696608550181, 8.378764887855950169310935047772, 9.021408372978754718610141754146, 9.805353329081282244670300954756
