| L(s) = 1 | + 7.05·2-s − 22.8·3-s + 17.7·4-s − 160.·6-s − 141.·7-s − 100.·8-s + 277.·9-s − 493.·11-s − 404.·12-s + 169·13-s − 996.·14-s − 1.27e3·16-s − 1.24e3·17-s + 1.95e3·18-s + 2.45e3·19-s + 3.22e3·21-s − 3.48e3·22-s − 2.62e3·23-s + 2.29e3·24-s + 1.19e3·26-s − 791.·27-s − 2.50e3·28-s − 4.50e3·29-s + 5.77e3·31-s − 5.78e3·32-s + 1.12e4·33-s − 8.78e3·34-s + ⋯ |
| L(s) = 1 | + 1.24·2-s − 1.46·3-s + 0.554·4-s − 1.82·6-s − 1.09·7-s − 0.555·8-s + 1.14·9-s − 1.23·11-s − 0.811·12-s + 0.277·13-s − 1.35·14-s − 1.24·16-s − 1.04·17-s + 1.42·18-s + 1.56·19-s + 1.59·21-s − 1.53·22-s − 1.03·23-s + 0.813·24-s + 0.345·26-s − 0.209·27-s − 0.604·28-s − 0.994·29-s + 1.08·31-s − 0.999·32-s + 1.80·33-s − 1.30·34-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
| \(L(3)\) |
\(\approx\) |
\(0.9081041463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9081041463\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 7.05T + 32T^{2} \) |
| 3 | \( 1 + 22.8T + 243T^{2} \) |
| 7 | \( 1 + 141.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 493.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.24e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.45e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.62e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.50e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.77e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.08e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.45e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.84e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.79e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.90e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.03e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.41e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.05e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.99e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.10e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.78e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.26e5T + 8.58e9T^{2} \) |
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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
−11.03651903350339869981650625994, −10.18371806933519546855430913719, −9.113605460383680327579578334382, −7.43071966009123120761384590713, −6.34861494697086571397659072215, −5.73486539038569856436923229436, −4.99031563794642622927811443719, −3.86860504976223299157370362557, −2.64595666002383962832282486320, −0.44491041399355830094882298927,
0.44491041399355830094882298927, 2.64595666002383962832282486320, 3.86860504976223299157370362557, 4.99031563794642622927811443719, 5.73486539038569856436923229436, 6.34861494697086571397659072215, 7.43071966009123120761384590713, 9.113605460383680327579578334382, 10.18371806933519546855430913719, 11.03651903350339869981650625994
