| L(s) = 1 | + 2.66·2-s + 3.11·3-s + 5.07·4-s − 5-s + 8.29·6-s − 0.107·7-s + 8.18·8-s + 6.73·9-s − 2.66·10-s + 11-s + 15.8·12-s − 1.25·13-s − 0.285·14-s − 3.11·15-s + 11.6·16-s − 0.853·17-s + 17.9·18-s + 0.172·19-s − 5.07·20-s − 0.334·21-s + 2.66·22-s − 4.20·23-s + 25.5·24-s + 25-s − 3.34·26-s + 11.6·27-s − 0.545·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 1.80·3-s + 2.53·4-s − 0.447·5-s + 3.38·6-s − 0.0405·7-s + 2.89·8-s + 2.24·9-s − 0.841·10-s + 0.301·11-s + 4.57·12-s − 0.349·13-s − 0.0763·14-s − 0.805·15-s + 2.90·16-s − 0.206·17-s + 4.22·18-s + 0.0395·19-s − 1.13·20-s − 0.0730·21-s + 0.567·22-s − 0.877·23-s + 5.21·24-s + 0.200·25-s − 0.656·26-s + 2.24·27-s − 0.102·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.34258779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.34258779\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 3 | \( 1 - 3.11T + 3T^{2} \) |
| 7 | \( 1 + 0.107T + 7T^{2} \) |
| 13 | \( 1 + 1.25T + 13T^{2} \) |
| 17 | \( 1 + 0.853T + 17T^{2} \) |
| 19 | \( 1 - 0.172T + 19T^{2} \) |
| 23 | \( 1 + 4.20T + 23T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 + 4.36T + 31T^{2} \) |
| 37 | \( 1 + 4.99T + 37T^{2} \) |
| 41 | \( 1 + 0.464T + 41T^{2} \) |
| 43 | \( 1 - 4.96T + 43T^{2} \) |
| 47 | \( 1 + 3.55T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 7.60T + 59T^{2} \) |
| 61 | \( 1 + 1.21T + 61T^{2} \) |
| 67 | \( 1 - 1.30T + 67T^{2} \) |
| 71 | \( 1 - 6.76T + 71T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 8.45T + 89T^{2} \) |
| 97 | \( 1 - 7.24T + 97T^{2} \) |
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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
−8.121683555934167234693713841561, −7.60580838889328205074827771333, −6.99735214452475099234644007836, −6.22065598387206644051837051736, −5.15283367710634652202923918672, −4.41526070617348525146494388069, −3.58495593236792092179925107601, −3.42835934575709513248469172895, −2.29898055858439455545412889499, −1.78654990665797863767408054607,
1.78654990665797863767408054607, 2.29898055858439455545412889499, 3.42835934575709513248469172895, 3.58495593236792092179925107601, 4.41526070617348525146494388069, 5.15283367710634652202923918672, 6.22065598387206644051837051736, 6.99735214452475099234644007836, 7.60580838889328205074827771333, 8.121683555934167234693713841561
