| L(s) = 1 | − 9.62·2-s − 10.7·3-s + 60.5·4-s + 103.·6-s − 18.4·7-s − 274.·8-s − 128.·9-s − 80.7·11-s − 648.·12-s + 169·13-s + 177.·14-s + 704.·16-s − 170.·17-s + 1.23e3·18-s + 722.·19-s + 197.·21-s + 777.·22-s + 519.·23-s + 2.94e3·24-s − 1.62e3·26-s + 3.97e3·27-s − 1.11e3·28-s + 2.39e3·29-s − 6.73e3·31-s + 2.01e3·32-s + 865.·33-s + 1.63e3·34-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.687·3-s + 1.89·4-s + 1.16·6-s − 0.142·7-s − 1.51·8-s − 0.527·9-s − 0.201·11-s − 1.30·12-s + 0.277·13-s + 0.241·14-s + 0.687·16-s − 0.142·17-s + 0.896·18-s + 0.459·19-s + 0.0977·21-s + 0.342·22-s + 0.204·23-s + 1.04·24-s − 0.471·26-s + 1.05·27-s − 0.268·28-s + 0.528·29-s − 1.25·31-s + 0.347·32-s + 0.138·33-s + 0.242·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3546111323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3546111323\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 + 9.62T + 32T^{2} \) |
| 3 | \( 1 + 10.7T + 243T^{2} \) |
| 7 | \( 1 + 18.4T + 1.68e4T^{2} \) |
| 11 | \( 1 + 80.7T + 1.61e5T^{2} \) |
| 17 | \( 1 + 170.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 722.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 519.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.52e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 855.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 553.T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 697.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.29e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.95e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.23e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.99e4T + 8.58e9T^{2} \) |
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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
−10.66840556252990131837048038287, −9.840990494340263420199321939359, −8.891557011162821417739681690781, −8.159429231670706229232040098433, −7.08408008482678491539028345402, −6.24042733968256219357825140825, −5.07534695243356204283293911182, −3.14400769858179058408582426565, −1.70407370363802893987987347740, −0.43909575249550549659992038078,
0.43909575249550549659992038078, 1.70407370363802893987987347740, 3.14400769858179058408582426565, 5.07534695243356204283293911182, 6.24042733968256219357825140825, 7.08408008482678491539028345402, 8.159429231670706229232040098433, 8.891557011162821417739681690781, 9.840990494340263420199321939359, 10.66840556252990131837048038287
