| L(s) = 1 | − 1.73·2-s + 2.71·3-s + 1.02·4-s − 1.00·5-s − 4.71·6-s + 1.91·7-s + 1.70·8-s + 4.35·9-s + 1.74·10-s + 11-s + 2.77·12-s + 6.51·13-s − 3.33·14-s − 2.72·15-s − 4.99·16-s + 17-s − 7.57·18-s − 0.645·19-s − 1.02·20-s + 5.20·21-s − 1.73·22-s + 4.75·23-s + 4.61·24-s − 3.99·25-s − 11.3·26-s + 3.68·27-s + 1.96·28-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 1.56·3-s + 0.510·4-s − 0.448·5-s − 1.92·6-s + 0.725·7-s + 0.601·8-s + 1.45·9-s + 0.551·10-s + 0.301·11-s + 0.799·12-s + 1.80·13-s − 0.891·14-s − 0.702·15-s − 1.24·16-s + 0.242·17-s − 1.78·18-s − 0.148·19-s − 0.229·20-s + 1.13·21-s − 0.370·22-s + 0.992·23-s + 0.941·24-s − 0.798·25-s − 2.22·26-s + 0.708·27-s + 0.370·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.396766003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.396766003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 - 2.71T + 3T^{2} \) |
| 5 | \( 1 + 1.00T + 5T^{2} \) |
| 7 | \( 1 - 1.91T + 7T^{2} \) |
| 13 | \( 1 - 6.51T + 13T^{2} \) |
| 19 | \( 1 + 0.645T + 19T^{2} \) |
| 23 | \( 1 - 4.75T + 23T^{2} \) |
| 29 | \( 1 - 6.97T + 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + 0.654T + 37T^{2} \) |
| 41 | \( 1 - 9.47T + 41T^{2} \) |
| 47 | \( 1 - 1.37T + 47T^{2} \) |
| 53 | \( 1 + 3.32T + 53T^{2} \) |
| 59 | \( 1 + 6.00T + 59T^{2} \) |
| 61 | \( 1 + 0.664T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 1.15T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 2.72T + 97T^{2} \) |
| 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
−8.170123989698544774822470389184, −7.60296107100694009438988238358, −6.84984619833263146501599627564, −6.00123052474397449973381217383, −4.70562014737472972146974937462, −4.12922284402859652213182864581, −3.39312811651496440651287504104, −2.52583241221422023757592134484, −1.53218719429846495416138695886, −0.975124882432195451160528620789,
0.975124882432195451160528620789, 1.53218719429846495416138695886, 2.52583241221422023757592134484, 3.39312811651496440651287504104, 4.12922284402859652213182864581, 4.70562014737472972146974937462, 6.00123052474397449973381217383, 6.84984619833263146501599627564, 7.60296107100694009438988238358, 8.170123989698544774822470389184
