| L(s) = 1 | + 4.19·2-s + 1.66·3-s + 9.56·4-s − 10.9·5-s + 6.99·6-s + 6.55·8-s − 24.2·9-s − 45.7·10-s + 20.6·11-s + 15.9·12-s − 41.1·13-s − 18.2·15-s − 49.0·16-s + 84.1·17-s − 101.·18-s + 67.9·19-s − 104.·20-s + 86.6·22-s + 52.1·23-s + 10.9·24-s − 5.86·25-s − 172.·26-s − 85.4·27-s + 44.2·29-s − 76.2·30-s + 107.·31-s − 257.·32-s + ⋯ |
| L(s) = 1 | + 1.48·2-s + 0.320·3-s + 1.19·4-s − 0.976·5-s + 0.475·6-s + 0.289·8-s − 0.896·9-s − 1.44·10-s + 0.566·11-s + 0.383·12-s − 0.878·13-s − 0.313·15-s − 0.766·16-s + 1.20·17-s − 1.32·18-s + 0.820·19-s − 1.16·20-s + 0.839·22-s + 0.472·23-s + 0.0930·24-s − 0.0469·25-s − 1.30·26-s − 0.608·27-s + 0.283·29-s − 0.464·30-s + 0.621·31-s − 1.42·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.063248494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.063248494\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 - 4.19T + 8T^{2} \) |
| 3 | \( 1 - 1.66T + 27T^{2} \) |
| 5 | \( 1 + 10.9T + 125T^{2} \) |
| 11 | \( 1 - 20.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 67.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 52.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 44.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 107.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 48.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 396.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 482.T + 7.95e4T^{2} \) |
| 53 | \( 1 - 746.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 249.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 247.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 580.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 18.2T + 3.57e5T^{2} \) |
| 73 | \( 1 - 214.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 877.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 322.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.65e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−8.524305932536478355447682958544, −7.68921376713798099163937014921, −7.06712222868980550996412664518, −6.06602838497693465219716600141, −5.33380626807347904326038562265, −4.63425300567686447371290749949, −3.66675716791817519390925987127, −3.22946252496919761438132140973, −2.33020013670465383325193898034, −0.68626784375664600672380632454,
0.68626784375664600672380632454, 2.33020013670465383325193898034, 3.22946252496919761438132140973, 3.66675716791817519390925987127, 4.63425300567686447371290749949, 5.33380626807347904326038562265, 6.06602838497693465219716600141, 7.06712222868980550996412664518, 7.68921376713798099163937014921, 8.524305932536478355447682958544
