| L(s) = 1 | − 3.81·2-s + 9·3-s − 17.4·4-s − 34.2·6-s + 89.4·7-s + 188.·8-s + 81·9-s − 121·11-s − 157.·12-s + 373.·13-s − 340.·14-s − 159.·16-s − 243.·17-s − 308.·18-s − 2.33e3·19-s + 804.·21-s + 461.·22-s + 3.30e3·23-s + 1.69e3·24-s − 1.42e3·26-s + 729·27-s − 1.56e3·28-s − 5.63e3·29-s − 1.22e3·31-s − 5.42e3·32-s − 1.08e3·33-s + 929.·34-s + ⋯ |
| L(s) = 1 | − 0.673·2-s + 0.577·3-s − 0.546·4-s − 0.388·6-s + 0.689·7-s + 1.04·8-s + 0.333·9-s − 0.301·11-s − 0.315·12-s + 0.613·13-s − 0.464·14-s − 0.155·16-s − 0.204·17-s − 0.224·18-s − 1.48·19-s + 0.398·21-s + 0.203·22-s + 1.30·23-s + 0.601·24-s − 0.413·26-s + 0.192·27-s − 0.376·28-s − 1.24·29-s − 0.228·31-s − 0.936·32-s − 0.174·33-s + 0.137·34-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 3.81T + 32T^{2} \) |
| 7 | \( 1 - 89.4T + 1.68e4T^{2} \) |
| 13 | \( 1 - 373.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 243.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.33e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.30e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.22e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.06e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.97e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.26e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.89e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.37e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.01e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.33e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.87e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.24e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.40e4T + 8.58e9T^{2} \) |
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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
−8.925616251281871092301075366575, −8.374341286470535156291761564720, −7.67674389833094828607570212684, −6.71106970935721035818375037207, −5.36146617799064621102707479534, −4.49137543744876686813502999855, −3.60053322091560238297153044617, −2.17424876243105158567107649420, −1.24749719484695342235048556838, 0,
1.24749719484695342235048556838, 2.17424876243105158567107649420, 3.60053322091560238297153044617, 4.49137543744876686813502999855, 5.36146617799064621102707479534, 6.71106970935721035818375037207, 7.67674389833094828607570212684, 8.374341286470535156291761564720, 8.925616251281871092301075366575
