| L(s) = 1 | + 2-s + 3.20·3-s + 4-s − 4.32·5-s + 3.20·6-s + 8-s + 7.24·9-s − 4.32·10-s + 1.98·11-s + 3.20·12-s − 5.18·13-s − 13.8·15-s + 16-s + 1.92·17-s + 7.24·18-s + 1.46·19-s − 4.32·20-s + 1.98·22-s + 5.00·23-s + 3.20·24-s + 13.7·25-s − 5.18·26-s + 13.6·27-s + 7.47·29-s − 13.8·30-s + 2.00·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.84·3-s + 0.5·4-s − 1.93·5-s + 1.30·6-s + 0.353·8-s + 2.41·9-s − 1.36·10-s + 0.598·11-s + 0.924·12-s − 1.43·13-s − 3.57·15-s + 0.250·16-s + 0.465·17-s + 1.70·18-s + 0.335·19-s − 0.967·20-s + 0.422·22-s + 1.04·23-s + 0.653·24-s + 2.74·25-s − 1.01·26-s + 2.61·27-s + 1.38·29-s − 2.53·30-s + 0.359·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.573087810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.573087810\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 3.20T + 3T^{2} \) |
| 5 | \( 1 + 4.32T + 5T^{2} \) |
| 11 | \( 1 - 1.98T + 11T^{2} \) |
| 13 | \( 1 + 5.18T + 13T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 - 5.00T + 23T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 - 2.00T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 43 | \( 1 + 3.82T + 43T^{2} \) |
| 47 | \( 1 + 2.75T + 47T^{2} \) |
| 53 | \( 1 + 8.10T + 53T^{2} \) |
| 59 | \( 1 + 0.763T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 3.26T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 2.91T + 73T^{2} \) |
| 79 | \( 1 + 2.10T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 3.97T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−8.206026085002605939933302558149, −7.78440535694896560021307489514, −7.17617162900307005900738083058, −6.67126949251585656436973279698, −4.85870298952730109354200784920, −4.53338661437501560027361334722, −3.66388192264967473794813552667, −3.13800707454505423964872492841, −2.51060707584940765521604818800, −1.05962810148594300175160918164,
1.05962810148594300175160918164, 2.51060707584940765521604818800, 3.13800707454505423964872492841, 3.66388192264967473794813552667, 4.53338661437501560027361334722, 4.85870298952730109354200784920, 6.67126949251585656436973279698, 7.17617162900307005900738083058, 7.78440535694896560021307489514, 8.206026085002605939933302558149
