| L(s) = 1 | − 1.00·2-s + 20.9·3-s − 30.9·4-s − 25·5-s − 20.9·6-s + 215.·7-s + 63.0·8-s + 194.·9-s + 25.0·10-s − 121·11-s − 648.·12-s + 358.·13-s − 215.·14-s − 522.·15-s + 928.·16-s + 493.·17-s − 194.·18-s + 361·19-s + 774.·20-s + 4.50e3·21-s + 121.·22-s + 4.06e3·23-s + 1.31e3·24-s + 625·25-s − 358.·26-s − 1.01e3·27-s − 6.67e3·28-s + ⋯ |
| L(s) = 1 | − 0.176·2-s + 1.34·3-s − 0.968·4-s − 0.447·5-s − 0.237·6-s + 1.66·7-s + 0.348·8-s + 0.799·9-s + 0.0791·10-s − 0.301·11-s − 1.29·12-s + 0.587·13-s − 0.293·14-s − 0.599·15-s + 0.907·16-s + 0.413·17-s − 0.141·18-s + 0.229·19-s + 0.433·20-s + 2.22·21-s + 0.0533·22-s + 1.60·23-s + 0.467·24-s + 0.200·25-s − 0.104·26-s − 0.269·27-s − 1.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.492820753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.492820753\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 1.00T + 32T^{2} \) |
| 3 | \( 1 - 20.9T + 243T^{2} \) |
| 7 | \( 1 - 215.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 358.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 493.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 4.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.19e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.91e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 216.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.47e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.23e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.82e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.22e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.88e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.97e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.38e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.34e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.08e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.19e4T + 8.58e9T^{2} \) |
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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
−8.859909417832128622314970538054, −8.405853223153962511870233780473, −7.929578332501481962870560090386, −7.14601691101553033240619774841, −5.40942580977686354030203830815, −4.75288665682391270777617921595, −3.84619474839713431632574236308, −2.99873821317279546671902106280, −1.71970326177562358864173103620, −0.840503953745633211165892675850,
0.840503953745633211165892675850, 1.71970326177562358864173103620, 2.99873821317279546671902106280, 3.84619474839713431632574236308, 4.75288665682391270777617921595, 5.40942580977686354030203830815, 7.14601691101553033240619774841, 7.929578332501481962870560090386, 8.405853223153962511870233780473, 8.859909417832128622314970538054
