| L(s) = 1 | + 2·2-s − 6.53·3-s + 4·4-s + 20.9·5-s − 13.0·6-s + 8.55·7-s + 8·8-s + 15.6·9-s + 41.8·10-s + 10.8·11-s − 26.1·12-s + 54.7·13-s + 17.1·14-s − 136.·15-s + 16·16-s − 106.·17-s + 31.2·18-s − 113.·19-s + 83.6·20-s − 55.8·21-s + 21.6·22-s − 112.·23-s − 52.2·24-s + 312.·25-s + 109.·26-s + 74.1·27-s + 34.2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.25·3-s + 0.5·4-s + 1.87·5-s − 0.888·6-s + 0.462·7-s + 0.353·8-s + 0.579·9-s + 1.32·10-s + 0.296·11-s − 0.628·12-s + 1.16·13-s + 0.326·14-s − 2.35·15-s + 0.250·16-s − 1.51·17-s + 0.409·18-s − 1.37·19-s + 0.935·20-s − 0.580·21-s + 0.209·22-s − 1.02·23-s − 0.444·24-s + 2.50·25-s + 0.825·26-s + 0.528·27-s + 0.231·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.885434255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.885434255\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 6.53T + 27T^{2} \) |
| 5 | \( 1 - 20.9T + 125T^{2} \) |
| 7 | \( 1 - 8.55T + 343T^{2} \) |
| 11 | \( 1 - 10.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 54.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 112.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 102.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 105.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 216.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 102.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 455.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 593.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 558.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 473.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 193.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 2.38T + 3.57e5T^{2} \) |
| 73 | \( 1 - 119.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 964.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 772.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.34e3T + 9.12e5T^{2} \) |
| 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
−14.37754822797944585925893777246, −13.49673397096277565047651673604, −12.57758249238211522831394727583, −11.12319808308446208598881573956, −10.53714188433198907932876274037, −8.914897700803803362109616469528, −6.39247721929765490332409662893, −6.02323675885339702628982660193, −4.67410619931427632141513525141, −1.88016279849672962719054285816,
1.88016279849672962719054285816, 4.67410619931427632141513525141, 6.02323675885339702628982660193, 6.39247721929765490332409662893, 8.914897700803803362109616469528, 10.53714188433198907932876274037, 11.12319808308446208598881573956, 12.57758249238211522831394727583, 13.49673397096277565047651673604, 14.37754822797944585925893777246
