| L(s) = 1 | − 2-s + 4-s − 0.725·5-s + 1.29·7-s − 8-s + 0.725·10-s + 3.71·11-s + 2.30·13-s − 1.29·14-s + 16-s − 6.32·17-s + 0.0212·19-s − 0.725·20-s − 3.71·22-s + 1.30·23-s − 4.47·25-s − 2.30·26-s + 1.29·28-s + 2.06·29-s − 5.82·31-s − 32-s + 6.32·34-s − 0.942·35-s − 7.25·37-s − 0.0212·38-s + 0.725·40-s − 8.77·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.324·5-s + 0.491·7-s − 0.353·8-s + 0.229·10-s + 1.12·11-s + 0.640·13-s − 0.347·14-s + 0.250·16-s − 1.53·17-s + 0.00487·19-s − 0.162·20-s − 0.792·22-s + 0.272·23-s − 0.894·25-s − 0.452·26-s + 0.245·28-s + 0.383·29-s − 1.04·31-s − 0.176·32-s + 1.08·34-s − 0.159·35-s − 1.19·37-s − 0.00344·38-s + 0.114·40-s − 1.37·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 269 | \( 1 - T \) |
| good | 5 | \( 1 + 0.725T + 5T^{2} \) |
| 7 | \( 1 - 1.29T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 + 6.32T + 17T^{2} \) |
| 19 | \( 1 - 0.0212T + 19T^{2} \) |
| 23 | \( 1 - 1.30T + 23T^{2} \) |
| 29 | \( 1 - 2.06T + 29T^{2} \) |
| 31 | \( 1 + 5.82T + 31T^{2} \) |
| 37 | \( 1 + 7.25T + 37T^{2} \) |
| 41 | \( 1 + 8.77T + 41T^{2} \) |
| 43 | \( 1 - 4.52T + 43T^{2} \) |
| 47 | \( 1 + 3.85T + 47T^{2} \) |
| 53 | \( 1 - 6.86T + 53T^{2} \) |
| 59 | \( 1 + 7.84T + 59T^{2} \) |
| 61 | \( 1 - 3.84T + 61T^{2} \) |
| 67 | \( 1 + 9.31T + 67T^{2} \) |
| 71 | \( 1 - 0.643T + 71T^{2} \) |
| 73 | \( 1 + 2.97T + 73T^{2} \) |
| 79 | \( 1 - 0.418T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 5.07T + 97T^{2} \) |
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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
−8.062127020476945639664024893784, −7.17954661043057910354999758143, −6.65764583378435721577569252726, −5.93920438482215233490347210947, −4.91532294570424518987677084412, −4.06808824151098825977657742569, −3.36635314222214616666820912028, −2.08396770372399139405525142067, −1.39065254770339010218440526589, 0,
1.39065254770339010218440526589, 2.08396770372399139405525142067, 3.36635314222214616666820912028, 4.06808824151098825977657742569, 4.91532294570424518987677084412, 5.93920438482215233490347210947, 6.65764583378435721577569252726, 7.17954661043057910354999758143, 8.062127020476945639664024893784
