| L(s) = 1 | − 2-s + 2.33·3-s + 4-s − 5-s − 2.33·6-s − 4.98·7-s − 8-s + 2.43·9-s + 10-s + 5.63·11-s + 2.33·12-s − 13-s + 4.98·14-s − 2.33·15-s + 16-s + 1.67·17-s − 2.43·18-s + 3.44·19-s − 20-s − 11.6·21-s − 5.63·22-s − 7.10·23-s − 2.33·24-s + 25-s + 26-s − 1.31·27-s − 4.98·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·3-s + 0.5·4-s − 0.447·5-s − 0.951·6-s − 1.88·7-s − 0.353·8-s + 0.811·9-s + 0.316·10-s + 1.69·11-s + 0.672·12-s − 0.277·13-s + 1.33·14-s − 0.601·15-s + 0.250·16-s + 0.405·17-s − 0.573·18-s + 0.790·19-s − 0.223·20-s − 2.53·21-s − 1.20·22-s − 1.48·23-s − 0.475·24-s + 0.200·25-s + 0.196·26-s − 0.253·27-s − 0.942·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.636452074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.636452074\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 2.33T + 3T^{2} \) |
| 7 | \( 1 + 4.98T + 7T^{2} \) |
| 11 | \( 1 - 5.63T + 11T^{2} \) |
| 17 | \( 1 - 1.67T + 17T^{2} \) |
| 19 | \( 1 - 3.44T + 19T^{2} \) |
| 23 | \( 1 + 7.10T + 23T^{2} \) |
| 29 | \( 1 + 4.27T + 29T^{2} \) |
| 37 | \( 1 - 8.84T + 37T^{2} \) |
| 41 | \( 1 - 9.84T + 41T^{2} \) |
| 43 | \( 1 + 8.68T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 8.95T + 53T^{2} \) |
| 59 | \( 1 + 4.59T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 0.980T + 67T^{2} \) |
| 71 | \( 1 + 2.38T + 71T^{2} \) |
| 73 | \( 1 - 6.60T + 73T^{2} \) |
| 79 | \( 1 - 0.835T + 79T^{2} \) |
| 83 | \( 1 + 0.912T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 3.87T + 97T^{2} \) |
| 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
−8.624147731519683588966156063287, −7.69431494896776579782594939348, −7.28850294822317086412583553990, −6.37744988117319321879814564573, −5.89408412592496413481939528486, −4.12750271605898360972305813139, −3.62614426280046518986642087558, −3.00372154453413772279774609686, −2.08901272163641145056779366758, −0.74853348840063989507661497198,
0.74853348840063989507661497198, 2.08901272163641145056779366758, 3.00372154453413772279774609686, 3.62614426280046518986642087558, 4.12750271605898360972305813139, 5.89408412592496413481939528486, 6.37744988117319321879814564573, 7.28850294822317086412583553990, 7.69431494896776579782594939348, 8.624147731519683588966156063287
