| L(s) = 1 | − 9.60·2-s + 25.9·3-s + 60.1·4-s − 248.·6-s + 181.·7-s − 270.·8-s + 428.·9-s + 512.·11-s + 1.56e3·12-s + 169·13-s − 1.74e3·14-s + 672.·16-s + 1.66e3·17-s − 4.11e3·18-s − 464.·19-s + 4.70e3·21-s − 4.92e3·22-s − 2.09e3·23-s − 7.01e3·24-s − 1.62e3·26-s + 4.82e3·27-s + 1.09e4·28-s + 5.93e3·29-s + 8.39e3·31-s + 2.20e3·32-s + 1.32e4·33-s − 1.60e4·34-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.66·3-s + 1.88·4-s − 2.82·6-s + 1.40·7-s − 1.49·8-s + 1.76·9-s + 1.27·11-s + 3.12·12-s + 0.277·13-s − 2.37·14-s + 0.656·16-s + 1.39·17-s − 2.99·18-s − 0.294·19-s + 2.32·21-s − 2.16·22-s − 0.825·23-s − 2.48·24-s − 0.470·26-s + 1.27·27-s + 2.63·28-s + 1.31·29-s + 1.56·31-s + 0.380·32-s + 2.12·33-s − 2.37·34-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
| \(L(3)\) |
\(\approx\) |
\(2.478053470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.478053470\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 9.60T + 32T^{2} \) |
| 3 | \( 1 - 25.9T + 243T^{2} \) |
| 7 | \( 1 - 181.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 512.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.66e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 464.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.39e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.90e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.25e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.95e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.94e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.14e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.85e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.86e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 208.T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.52e4T + 8.58e9T^{2} \) |
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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
−10.27421448342698225035419693156, −9.635809057614581006594180955082, −8.659535395098699533700929676775, −8.232163248035386723699358428147, −7.63407602557154908753830264917, −6.48605422770949239125784724729, −4.44630299701614163657500176159, −3.08253586340735132293688363457, −1.76085979538012763996660744846, −1.21839393479287501286364824399,
1.21839393479287501286364824399, 1.76085979538012763996660744846, 3.08253586340735132293688363457, 4.44630299701614163657500176159, 6.48605422770949239125784724729, 7.63407602557154908753830264917, 8.232163248035386723699358428147, 8.659535395098699533700929676775, 9.635809057614581006594180955082, 10.27421448342698225035419693156
