| L(s) = 1 | − 5.91·2-s + 18.2·3-s + 3.00·4-s − 108.·6-s − 139.·7-s + 171.·8-s + 91.2·9-s + 533.·11-s + 54.9·12-s + 675.·13-s + 825.·14-s − 1.11e3·16-s + 268.·17-s − 539.·18-s − 2.64e3·19-s − 2.55e3·21-s − 3.15e3·22-s − 794.·23-s + 3.13e3·24-s − 3.99e3·26-s − 2.77e3·27-s − 419.·28-s − 841·29-s − 4.23e3·31-s + 1.08e3·32-s + 9.74e3·33-s − 1.59e3·34-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 1.17·3-s + 0.0939·4-s − 1.22·6-s − 1.07·7-s + 0.947·8-s + 0.375·9-s + 1.32·11-s + 0.110·12-s + 1.10·13-s + 1.12·14-s − 1.08·16-s + 0.225·17-s − 0.392·18-s − 1.68·19-s − 1.26·21-s − 1.38·22-s − 0.313·23-s + 1.11·24-s − 1.16·26-s − 0.732·27-s − 0.101·28-s − 0.185·29-s − 0.790·31-s + 0.187·32-s + 1.55·33-s − 0.236·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 + 841T \) |
| good | 2 | \( 1 + 5.91T + 32T^{2} \) |
| 3 | \( 1 - 18.2T + 243T^{2} \) |
| 7 | \( 1 + 139.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 533.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 675.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 268.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.64e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 794.T + 6.43e6T^{2} \) |
| 31 | \( 1 + 4.23e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.68e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.38e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.12e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.39e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.78e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.57e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.99e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.99e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.38e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.30e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.38e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.76e5T + 8.58e9T^{2} \) |
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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
−9.060009551564571432351223268928, −8.669154369141142042850108573675, −7.83222640533842914913579009437, −6.79130130744968867734248190191, −5.99730056868086546160730486841, −4.13049943542488389684218751349, −3.63564361803997121331456573760, −2.31003280874632510271838211059, −1.24920773931922211368448935562, 0,
1.24920773931922211368448935562, 2.31003280874632510271838211059, 3.63564361803997121331456573760, 4.13049943542488389684218751349, 5.99730056868086546160730486841, 6.79130130744968867734248190191, 7.83222640533842914913579009437, 8.669154369141142042850108573675, 9.060009551564571432351223268928
