| L(s) = 1 | + 2·2-s + 4·4-s − 19.8·5-s + 7·7-s + 8·8-s − 39.7·10-s − 39.7·11-s + 88.6·13-s + 14·14-s + 16·16-s + 72.7·17-s + 38·19-s − 79.4·20-s − 79.4·22-s + 7.12·23-s + 269.·25-s + 177.·26-s + 28·28-s + 257.·29-s − 48.6·31-s + 32·32-s + 145.·34-s − 139.·35-s − 343.·37-s + 76·38-s − 158.·40-s + 217.·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.77·5-s + 0.377·7-s + 0.353·8-s − 1.25·10-s − 1.08·11-s + 1.89·13-s + 0.267·14-s + 0.250·16-s + 1.03·17-s + 0.458·19-s − 0.888·20-s − 0.770·22-s + 0.0646·23-s + 2.15·25-s + 1.33·26-s + 0.188·28-s + 1.64·29-s − 0.281·31-s + 0.176·32-s + 0.733·34-s − 0.671·35-s − 1.52·37-s + 0.324·38-s − 0.628·40-s + 0.826·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.321567249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.321567249\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 19.8T + 125T^{2} \) |
| 11 | \( 1 + 39.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 88.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 38T + 6.85e3T^{2} \) |
| 23 | \( 1 - 7.12T + 1.21e4T^{2} \) |
| 29 | \( 1 - 257.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 48.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 343.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 217.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 447.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 387.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 322.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 539.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 303.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 365.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 615.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 123.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 399.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 633.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 823.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 607.T + 9.12e5T^{2} \) |
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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
−10.97165451761081610355686279421, −10.59292998865008419399015124764, −8.746887026393357270301910279798, −7.963416328527012659560659302271, −7.34491007101857132748352108514, −5.98996425800598358099339701706, −4.86491792565251425845499135750, −3.85213835253619089502171907284, −3.05136733157141973903684438154, −0.946873441645070368289427119632,
0.946873441645070368289427119632, 3.05136733157141973903684438154, 3.85213835253619089502171907284, 4.86491792565251425845499135750, 5.98996425800598358099339701706, 7.34491007101857132748352108514, 7.963416328527012659560659302271, 8.746887026393357270301910279798, 10.59292998865008419399015124764, 10.97165451761081610355686279421
