| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.554748720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.554748720\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 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
−9.152890717034264962529401790274, −8.083646232368286766443526519432, −7.953547314755208208763297026766, −6.60157718647180174128354158549, −5.84715882507690383860337300663, −5.12429129240387157418642259990, −3.45697913482976929063210122811, −2.79280088108981441409805346564, −1.75588742053450884896975315195, −1.23037882850054152441558727189,
1.23037882850054152441558727189, 1.75588742053450884896975315195, 2.79280088108981441409805346564, 3.45697913482976929063210122811, 5.12429129240387157418642259990, 5.84715882507690383860337300663, 6.60157718647180174128354158549, 7.953547314755208208763297026766, 8.083646232368286766443526519432, 9.152890717034264962529401790274
