| L(s) = 1 | − 2.22·2-s + 6.66·3-s − 3.06·4-s + 5·5-s − 14.8·6-s − 18.2·7-s + 24.5·8-s + 17.4·9-s − 11.1·10-s − 11·11-s − 20.4·12-s + 18.1·13-s + 40.5·14-s + 33.3·15-s − 30.1·16-s − 33.1·17-s − 38.7·18-s − 19·19-s − 15.3·20-s − 121.·21-s + 24.4·22-s + 195.·23-s + 163.·24-s + 25·25-s − 40.3·26-s − 63.7·27-s + 55.9·28-s + ⋯ |
| L(s) = 1 | − 0.785·2-s + 1.28·3-s − 0.382·4-s + 0.447·5-s − 1.00·6-s − 0.985·7-s + 1.08·8-s + 0.645·9-s − 0.351·10-s − 0.301·11-s − 0.491·12-s + 0.387·13-s + 0.774·14-s + 0.573·15-s − 0.470·16-s − 0.472·17-s − 0.507·18-s − 0.229·19-s − 0.171·20-s − 1.26·21-s + 0.236·22-s + 1.77·23-s + 1.39·24-s + 0.200·25-s − 0.304·26-s − 0.454·27-s + 0.377·28-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 | \( 1 + 2.22T + 8T^{2} \) |
| 3 | \( 1 - 6.66T + 27T^{2} \) |
| 7 | \( 1 + 18.2T + 343T^{2} \) |
| 13 | \( 1 - 18.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 33.1T + 4.91e3T^{2} \) |
| 23 | \( 1 - 195.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 60.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 314.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 95.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 129.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 415.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 279.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 508.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 15.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.06e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 199.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 541.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 826.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 748.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 231.T + 9.12e5T^{2} \) |
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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
−9.151857610784532252082599021691, −8.619950071648812838811363104671, −7.67247954808717272030798357807, −6.94406595332588300873178309156, −5.77692053965903822588308268652, −4.56577668696530596675253087096, −3.47719395435600555173850771477, −2.64723317665409908576776514244, −1.43440442004631384387216036690, 0,
1.43440442004631384387216036690, 2.64723317665409908576776514244, 3.47719395435600555173850771477, 4.56577668696530596675253087096, 5.77692053965903822588308268652, 6.94406595332588300873178309156, 7.67247954808717272030798357807, 8.619950071648812838811363104671, 9.151857610784532252082599021691
