| L(s) = 1 | − 2-s + 0.589·3-s + 4-s − 3.50·5-s − 0.589·6-s − 2.22·7-s − 8-s − 2.65·9-s + 3.50·10-s + 11-s + 0.589·12-s − 0.880·13-s + 2.22·14-s − 2.06·15-s + 16-s + 6.14·17-s + 2.65·18-s − 3.50·20-s − 1.31·21-s − 22-s − 2.19·23-s − 0.589·24-s + 7.31·25-s + 0.880·26-s − 3.33·27-s − 2.22·28-s − 2.43·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.340·3-s + 0.5·4-s − 1.56·5-s − 0.240·6-s − 0.840·7-s − 0.353·8-s − 0.884·9-s + 1.10·10-s + 0.301·11-s + 0.170·12-s − 0.244·13-s + 0.594·14-s − 0.534·15-s + 0.250·16-s + 1.49·17-s + 0.625·18-s − 0.784·20-s − 0.286·21-s − 0.213·22-s − 0.457·23-s − 0.120·24-s + 1.46·25-s + 0.172·26-s − 0.641·27-s − 0.420·28-s − 0.451·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3279211841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3279211841\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.589T + 3T^{2} \) |
| 5 | \( 1 + 3.50T + 5T^{2} \) |
| 7 | \( 1 + 2.22T + 7T^{2} \) |
| 13 | \( 1 + 0.880T + 13T^{2} \) |
| 17 | \( 1 - 6.14T + 17T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 + 2.43T + 29T^{2} \) |
| 31 | \( 1 + 3.54T + 31T^{2} \) |
| 37 | \( 1 - 0.730T + 37T^{2} \) |
| 41 | \( 1 + 9.08T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 8.41T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 8.96T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 + 5.04T + 67T^{2} \) |
| 71 | \( 1 + 8.55T + 71T^{2} \) |
| 73 | \( 1 + 6.37T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 6.55T + 97T^{2} \) |
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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
−7.903576170050393821278167141593, −7.35621943226197556938752505126, −6.72505655059218536874126420665, −5.87221152221100534967396266719, −5.10206584790432872034985863001, −3.95023068339566366770375952392, −3.35324674377750242687256999330, −2.95391547321472452125020916883, −1.62293295418966383700710624649, −0.30528402928927473362800048543,
0.30528402928927473362800048543, 1.62293295418966383700710624649, 2.95391547321472452125020916883, 3.35324674377750242687256999330, 3.95023068339566366770375952392, 5.10206584790432872034985863001, 5.87221152221100534967396266719, 6.72505655059218536874126420665, 7.35621943226197556938752505126, 7.903576170050393821278167141593
