| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 2·7-s + 8-s + 9-s + 10-s + 6.46·11-s − 12-s + 2·14-s − 15-s + 16-s + 4·17-s + 18-s + 7.46·19-s + 20-s − 2·21-s + 6.46·22-s − 3.73·23-s − 24-s + 25-s − 27-s + 2·28-s + 0.267·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.94·11-s − 0.288·12-s + 0.534·14-s − 0.258·15-s + 0.250·16-s + 0.970·17-s + 0.235·18-s + 1.71·19-s + 0.223·20-s − 0.436·21-s + 1.37·22-s − 0.778·23-s − 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.377·28-s + 0.0497·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.967137851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.967137851\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 6.46T + 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 7.46T + 19T^{2} \) |
| 23 | \( 1 + 3.73T + 23T^{2} \) |
| 29 | \( 1 - 0.267T + 29T^{2} \) |
| 31 | \( 1 + 1.73T + 31T^{2} \) |
| 37 | \( 1 - 9.19T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 - 3.53T + 47T^{2} \) |
| 53 | \( 1 - 0.928T + 53T^{2} \) |
| 59 | \( 1 + 8.46T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 8.92T + 83T^{2} \) |
| 89 | \( 1 + 0.535T + 89T^{2} \) |
| 97 | \( 1 - 0.535T + 97T^{2} \) |
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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
−8.005672680325393034875250680343, −7.39094291440003788623537522825, −6.56802889272984449570145842802, −5.99397028643759570584673653068, −5.33879163509771435445177954684, −4.61923272183024708031834371592, −3.84686925210050827359791778493, −3.05891814409991462635988419698, −1.64856487248706250236099508461, −1.19024479396772255097112433846,
1.19024479396772255097112433846, 1.64856487248706250236099508461, 3.05891814409991462635988419698, 3.84686925210050827359791778493, 4.61923272183024708031834371592, 5.33879163509771435445177954684, 5.99397028643759570584673653068, 6.56802889272984449570145842802, 7.39094291440003788623537522825, 8.005672680325393034875250680343
