| L(s) = 1 | + 1.13·2-s − 7.07·3-s − 6.71·4-s − 5·5-s − 8.02·6-s + 20.7·7-s − 16.7·8-s + 23.0·9-s − 5.67·10-s − 11·11-s + 47.4·12-s − 58.6·13-s + 23.5·14-s + 35.3·15-s + 34.7·16-s − 15.5·17-s + 26.1·18-s + 19·19-s + 33.5·20-s − 146.·21-s − 12.4·22-s + 81.0·23-s + 118.·24-s + 25·25-s − 66.5·26-s + 28.2·27-s − 139.·28-s + ⋯ |
| L(s) = 1 | + 0.401·2-s − 1.36·3-s − 0.838·4-s − 0.447·5-s − 0.546·6-s + 1.12·7-s − 0.738·8-s + 0.852·9-s − 0.179·10-s − 0.301·11-s + 1.14·12-s − 1.25·13-s + 0.449·14-s + 0.608·15-s + 0.542·16-s − 0.221·17-s + 0.342·18-s + 0.229·19-s + 0.375·20-s − 1.52·21-s − 0.121·22-s + 0.734·23-s + 1.00·24-s + 0.200·25-s − 0.502·26-s + 0.201·27-s − 0.939·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 1.13T + 8T^{2} \) |
| 3 | \( 1 + 7.07T + 27T^{2} \) |
| 7 | \( 1 - 20.7T + 343T^{2} \) |
| 13 | \( 1 + 58.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.5T + 4.91e3T^{2} \) |
| 23 | \( 1 - 81.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 107.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 297.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 310.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 54.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 49.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 322.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 467.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 161.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 258.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 38.0T + 3.00e5T^{2} \) |
| 71 | \( 1 + 408.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 811.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 99.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 380.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 287.T + 9.12e5T^{2} \) |
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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
−9.198557825005543752469269787558, −8.157971954632620685019513716198, −7.46769832671192553825617233536, −6.33117573848135033595848399652, −5.38132879734752445863455844420, −4.81309804972318618598425571090, −4.31215098177003608452690369513, −2.76742373907731929352417242941, −1.01912746365397939560395190336, 0,
1.01912746365397939560395190336, 2.76742373907731929352417242941, 4.31215098177003608452690369513, 4.81309804972318618598425571090, 5.38132879734752445863455844420, 6.33117573848135033595848399652, 7.46769832671192553825617233536, 8.157971954632620685019513716198, 9.198557825005543752469269787558
