| L(s) = 1 | + 10.8·2-s − 9·3-s + 85.3·4-s − 97.4·6-s + 188.·7-s + 578.·8-s + 81·9-s + 121·11-s − 768.·12-s − 1.06e3·13-s + 2.04e3·14-s + 3.53e3·16-s + 2.03e3·17-s + 877.·18-s + 285.·19-s − 1.69e3·21-s + 1.31e3·22-s + 589.·23-s − 5.20e3·24-s − 1.15e4·26-s − 729·27-s + 1.60e4·28-s + 5.68e3·29-s + 1.89e3·31-s + 1.97e4·32-s − 1.08e3·33-s + 2.20e4·34-s + ⋯ |
| L(s) = 1 | + 1.91·2-s − 0.577·3-s + 2.66·4-s − 1.10·6-s + 1.45·7-s + 3.19·8-s + 0.333·9-s + 0.301·11-s − 1.54·12-s − 1.74·13-s + 2.78·14-s + 3.44·16-s + 1.70·17-s + 0.638·18-s + 0.181·19-s − 0.839·21-s + 0.577·22-s + 0.232·23-s − 1.84·24-s − 3.33·26-s − 0.192·27-s + 3.87·28-s + 1.25·29-s + 0.354·31-s + 3.41·32-s − 0.174·33-s + 3.26·34-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
| \(L(3)\) |
\(\approx\) |
\(9.143305900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.143305900\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 10.8T + 32T^{2} \) |
| 7 | \( 1 - 188.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.06e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 285.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 589.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.68e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.89e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.51e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.75e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.94e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.80e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 360.T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.27e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.22e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.78e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.80e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.15e5T + 8.58e9T^{2} \) |
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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
−9.992685797947536569340974896942, −8.173709209845834998290592945984, −7.38051637349913533573536072556, −6.69664019607838056070043152721, −5.42607783619281064149267181292, −5.11895241136977346503863795785, −4.40237629093841735261332313208, −3.24698809213074301522764452538, −2.14414999482247910137127548560, −1.15848924112631424389739746391,
1.15848924112631424389739746391, 2.14414999482247910137127548560, 3.24698809213074301522764452538, 4.40237629093841735261332313208, 5.11895241136977346503863795785, 5.42607783619281064149267181292, 6.69664019607838056070043152721, 7.38051637349913533573536072556, 8.173709209845834998290592945984, 9.992685797947536569340974896942
