L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 0.267·7-s + 8-s + 9-s − 10-s + 12-s − 4.46·13-s − 0.267·14-s − 15-s + 16-s − 1.73·17-s + 18-s − 7·19-s − 20-s − 0.267·21-s − 6.73·23-s + 24-s + 25-s − 4.46·26-s + 27-s − 0.267·28-s − 5·29-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 − 0.101·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s − 1.23·13-s − 0.0716·14-s − 0.258·15-s + 0.250·16-s − 0.420·17-s + 0.235·18-s − 1.60·19-s − 0.223·20-s − 0.0584·21-s − 1.40·23-s + 0.204·24-s + 0.200·25-s − 0.875·26-s + 0.192·27-s − 0.0506·28-s − 0.928·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + 0.267T + 7T^{2} \) |
| 13 | \( 1 + 4.46T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 + 6.73T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 6.19T + 31T^{2} \) |
| 37 | \( 1 + 7.19T + 37T^{2} \) |
| 41 | \( 1 - 4.19T + 41T^{2} \) |
| 43 | \( 1 - 2.19T + 43T^{2} \) |
| 47 | \( 1 - 4.92T + 47T^{2} \) |
| 53 | \( 1 - 0.732T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 - 4.53T + 67T^{2} \) |
| 71 | \( 1 + 2.46T + 71T^{2} \) |
| 73 | \( 1 - 9.66T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 9.53T + 83T^{2} \) |
| 89 | \( 1 - 9.46T + 89T^{2} \) |
| 97 | \( 1 - 8.73T + 97T^{2} \) |
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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
−7.987380278764521617704016378298, −7.51653426498253516174255243545, −6.62767653326035805056188088007, −6.01112544186951641746073845838, −4.85414793444078759597647877069, −4.32802134742402898192968693657, −3.56200413906123759278691398321, −2.55476549388887777358395752590, −1.89630619936456074985842817717, 0,
1.89630619936456074985842817717, 2.55476549388887777358395752590, 3.56200413906123759278691398321, 4.32802134742402898192968693657, 4.85414793444078759597647877069, 6.01112544186951641746073845838, 6.62767653326035805056188088007, 7.51653426498253516174255243545, 7.987380278764521617704016378298