| L(s) = 1 | − 1.08·2-s + 9·3-s − 30.8·4-s − 9.73·6-s + 139.·7-s + 67.9·8-s + 81·9-s − 121·11-s − 277.·12-s + 646.·13-s − 150.·14-s + 913.·16-s + 1.37e3·17-s − 87.5·18-s + 1.90e3·19-s + 1.25e3·21-s + 130.·22-s − 343.·23-s + 611.·24-s − 698.·26-s + 729·27-s − 4.30e3·28-s + 53.5·29-s + 634.·31-s − 3.16e3·32-s − 1.08e3·33-s − 1.49e3·34-s + ⋯ |
| L(s) = 1 | − 0.191·2-s + 0.577·3-s − 0.963·4-s − 0.110·6-s + 1.07·7-s + 0.375·8-s + 0.333·9-s − 0.301·11-s − 0.556·12-s + 1.06·13-s − 0.205·14-s + 0.891·16-s + 1.15·17-s − 0.0637·18-s + 1.21·19-s + 0.621·21-s + 0.0576·22-s − 0.135·23-s + 0.216·24-s − 0.202·26-s + 0.192·27-s − 1.03·28-s + 0.0118·29-s + 0.118·31-s − 0.545·32-s − 0.174·33-s − 0.221·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.781540747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.781540747\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 1.08T + 32T^{2} \) |
| 7 | \( 1 - 139.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 646.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.37e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.90e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 343.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 53.5T + 2.05e7T^{2} \) |
| 31 | \( 1 - 634.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.16e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.88e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.33e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.25e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.90e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.60e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 883.T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.39e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.17e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.58e5T + 8.58e9T^{2} \) |
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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
−9.404342941132456143639012171175, −8.482215120775309968013891966663, −8.057116307075131438285682697028, −7.25510354805086355741050604983, −5.70828310327807472909894539110, −5.03701218988820549792891851670, −3.99961804413794600935408905284, −3.16457992086396753472234850663, −1.62034891384160577330339635658, −0.843343357606116739670221804741,
0.843343357606116739670221804741, 1.62034891384160577330339635658, 3.16457992086396753472234850663, 3.99961804413794600935408905284, 5.03701218988820549792891851670, 5.70828310327807472909894539110, 7.25510354805086355741050604983, 8.057116307075131438285682697028, 8.482215120775309968013891966663, 9.404342941132456143639012171175
