| L(s) = 1 | + 5.23·2-s + 3·3-s + 19.4·4-s + 8.78·5-s + 15.7·6-s − 7·7-s + 60.0·8-s + 9·9-s + 46.0·10-s + 60.0·11-s + 58.3·12-s − 36.5·13-s − 36.6·14-s + 26.3·15-s + 158.·16-s − 91.3·17-s + 47.1·18-s − 98.2·19-s + 170.·20-s − 21·21-s + 314.·22-s + 23·23-s + 180.·24-s − 47.7·25-s − 191.·26-s + 27·27-s − 136.·28-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 0.577·3-s + 2.43·4-s + 0.786·5-s + 1.06·6-s − 0.377·7-s + 2.65·8-s + 0.333·9-s + 1.45·10-s + 1.64·11-s + 1.40·12-s − 0.779·13-s − 0.700·14-s + 0.453·15-s + 2.48·16-s − 1.30·17-s + 0.617·18-s − 1.18·19-s + 1.91·20-s − 0.218·21-s + 3.05·22-s + 0.208·23-s + 1.53·24-s − 0.382·25-s − 1.44·26-s + 0.192·27-s − 0.919·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.354850959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.354850959\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 5.23T + 8T^{2} \) |
| 5 | \( 1 - 8.78T + 125T^{2} \) |
| 11 | \( 1 - 60.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 36.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 91.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 98.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 260.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 238.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 222.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 104.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 314.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 235.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 145.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 42.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 81.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 74.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 447.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 926.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 103.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 275.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 943.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 8.45T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.88160796136026707284912952490, −9.728211723623793348239269953359, −8.901960259735109874545638917314, −7.34062575330067826888780156097, −6.49061547184769563512547811562, −5.94322567408653067644549057871, −4.53126360517461385104606697555, −3.95017979115012562015066611032, −2.64118856946265399453764663065, −1.82793202414803880947144036881,
1.82793202414803880947144036881, 2.64118856946265399453764663065, 3.95017979115012562015066611032, 4.53126360517461385104606697555, 5.94322567408653067644549057871, 6.49061547184769563512547811562, 7.34062575330067826888780156097, 8.901960259735109874545638917314, 9.728211723623793348239269953359, 10.88160796136026707284912952490
