| L(s) = 1 | − 2.55·2-s − 3-s + 4.51·4-s + 1.95·5-s + 2.55·6-s − 6.42·8-s + 9-s − 4.97·10-s + 5.66·11-s − 4.51·12-s + 4.68·13-s − 1.95·15-s + 7.36·16-s + 4.34·17-s − 2.55·18-s − 5.53·19-s + 8.80·20-s − 14.4·22-s − 3.74·23-s + 6.42·24-s − 1.19·25-s − 11.9·26-s − 27-s − 0.177·29-s + 4.97·30-s + 0.750·31-s − 5.96·32-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 0.577·3-s + 2.25·4-s + 0.872·5-s + 1.04·6-s − 2.27·8-s + 0.333·9-s − 1.57·10-s + 1.70·11-s − 1.30·12-s + 1.29·13-s − 0.503·15-s + 1.84·16-s + 1.05·17-s − 0.601·18-s − 1.27·19-s + 1.96·20-s − 3.08·22-s − 0.781·23-s + 1.31·24-s − 0.239·25-s − 2.34·26-s − 0.192·27-s − 0.0329·29-s + 0.908·30-s + 0.134·31-s − 1.05·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.038464109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.038464109\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 5 | \( 1 - 1.95T + 5T^{2} \) |
| 11 | \( 1 - 5.66T + 11T^{2} \) |
| 13 | \( 1 - 4.68T + 13T^{2} \) |
| 17 | \( 1 - 4.34T + 17T^{2} \) |
| 19 | \( 1 + 5.53T + 19T^{2} \) |
| 23 | \( 1 + 3.74T + 23T^{2} \) |
| 29 | \( 1 + 0.177T + 29T^{2} \) |
| 31 | \( 1 - 0.750T + 31T^{2} \) |
| 37 | \( 1 - 8.28T + 37T^{2} \) |
| 43 | \( 1 - 6.12T + 43T^{2} \) |
| 47 | \( 1 - 0.738T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 + 4.83T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 9.26T + 73T^{2} \) |
| 79 | \( 1 - 8.55T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 6.24T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.163365575174473010162460232949, −7.57725712092583355226525859158, −6.49854713480938420663771947000, −6.27339919038120426784903505772, −5.80976781635354601808759596888, −4.36212263246866663124636282513, −3.49740872547204308272411799596, −2.17356379870851892333469598356, −1.49213654519039066407273108895, −0.800906443967743727799047254780,
0.800906443967743727799047254780, 1.49213654519039066407273108895, 2.17356379870851892333469598356, 3.49740872547204308272411799596, 4.36212263246866663124636282513, 5.80976781635354601808759596888, 6.27339919038120426784903505772, 6.49854713480938420663771947000, 7.57725712092583355226525859158, 8.163365575174473010162460232949
