| L(s) = 1 | − 4.97·2-s − 9·3-s − 7.25·4-s + 44.7·6-s + 155.·7-s + 195.·8-s + 81·9-s + 121·11-s + 65.3·12-s + 1.17e3·13-s − 773.·14-s − 738.·16-s + 898.·17-s − 402.·18-s − 2.40e3·19-s − 1.39e3·21-s − 601.·22-s − 4.12e3·23-s − 1.75e3·24-s − 5.83e3·26-s − 729·27-s − 1.12e3·28-s + 75.4·29-s + 3.59e3·31-s − 2.57e3·32-s − 1.08e3·33-s − 4.46e3·34-s + ⋯ |
| L(s) = 1 | − 0.879·2-s − 0.577·3-s − 0.226·4-s + 0.507·6-s + 1.19·7-s + 1.07·8-s + 0.333·9-s + 0.301·11-s + 0.130·12-s + 1.92·13-s − 1.05·14-s − 0.721·16-s + 0.753·17-s − 0.293·18-s − 1.53·19-s − 0.692·21-s − 0.265·22-s − 1.62·23-s − 0.622·24-s − 1.69·26-s − 0.192·27-s − 0.271·28-s + 0.0166·29-s + 0.671·31-s − 0.444·32-s − 0.174·33-s − 0.662·34-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 | \( 1 + 4.97T + 32T^{2} \) |
| 7 | \( 1 - 155.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.17e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 898.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.40e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 75.4T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.60e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.50e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.28e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.69e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.28e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 882.T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.87e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.42e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.86e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.34e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.09e4T + 8.58e9T^{2} \) |
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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
−8.733397225383831556749220023170, −8.447036001593081185276902160881, −7.63808799092086733210418500688, −6.44418692752125496453718551220, −5.61334441679116164845507889151, −4.49115241417188941938272327159, −3.82997030810107324630633790480, −1.79094452225869315881115549246, −1.21187148122159902881890256635, 0,
1.21187148122159902881890256635, 1.79094452225869315881115549246, 3.82997030810107324630633790480, 4.49115241417188941938272327159, 5.61334441679116164845507889151, 6.44418692752125496453718551220, 7.63808799092086733210418500688, 8.447036001593081185276902160881, 8.733397225383831556749220023170
