| L(s) = 1 | − 2-s + 3-s + 4-s + 1.23·5-s − 6-s − 1.61·7-s − 8-s + 9-s − 1.23·10-s − 0.763·11-s + 12-s − 3.85·13-s + 1.61·14-s + 1.23·15-s + 16-s + 1.23·17-s − 18-s + 0.618·19-s + 1.23·20-s − 1.61·21-s + 0.763·22-s − 4.47·23-s − 24-s − 3.47·25-s + 3.85·26-s + 27-s − 1.61·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.552·5-s − 0.408·6-s − 0.611·7-s − 0.353·8-s + 0.333·9-s − 0.390·10-s − 0.230·11-s + 0.288·12-s − 1.06·13-s + 0.432·14-s + 0.319·15-s + 0.250·16-s + 0.299·17-s − 0.235·18-s + 0.141·19-s + 0.276·20-s − 0.353·21-s + 0.162·22-s − 0.932·23-s − 0.204·24-s − 0.694·25-s + 0.755·26-s + 0.192·27-s − 0.305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 + 3.85T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 - 0.618T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 0.472T + 41T^{2} \) |
| 43 | \( 1 - 0.381T + 43T^{2} \) |
| 47 | \( 1 + 1.70T + 47T^{2} \) |
| 53 | \( 1 + 0.763T + 53T^{2} \) |
| 59 | \( 1 + 6.76T + 59T^{2} \) |
| 61 | \( 1 + 1.09T + 61T^{2} \) |
| 67 | \( 1 - 3.38T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 2.61T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 + 3.09T + 97T^{2} \) |
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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
−7.905934469113832378847682768809, −7.21117575846749115427774350301, −6.41970328860297518514329823571, −5.83617341376181545776366604311, −4.86585656488935193425799695504, −3.94475216554488954829426898093, −2.87796836306277585670592466688, −2.41660207313527065507978661543, −1.37581589487773963178203556102, 0,
1.37581589487773963178203556102, 2.41660207313527065507978661543, 2.87796836306277585670592466688, 3.94475216554488954829426898093, 4.86585656488935193425799695504, 5.83617341376181545776366604311, 6.41970328860297518514329823571, 7.21117575846749115427774350301, 7.905934469113832378847682768809
