| L(s) = 1 | − 0.826·2-s − 3-s − 1.31·4-s − 3.86·5-s + 0.826·6-s + 7-s + 2.74·8-s + 9-s + 3.19·10-s − 5.36·11-s + 1.31·12-s − 5.44·13-s − 0.826·14-s + 3.86·15-s + 0.367·16-s + 3.56·17-s − 0.826·18-s − 0.606·19-s + 5.09·20-s − 21-s + 4.43·22-s + 3.70·23-s − 2.74·24-s + 9.94·25-s + 4.49·26-s − 27-s − 1.31·28-s + ⋯ |
| L(s) = 1 | − 0.584·2-s − 0.577·3-s − 0.658·4-s − 1.72·5-s + 0.337·6-s + 0.377·7-s + 0.969·8-s + 0.333·9-s + 1.01·10-s − 1.61·11-s + 0.380·12-s − 1.50·13-s − 0.220·14-s + 0.998·15-s + 0.0919·16-s + 0.863·17-s − 0.194·18-s − 0.139·19-s + 1.13·20-s − 0.218·21-s + 0.946·22-s + 0.771·23-s − 0.559·24-s + 1.98·25-s + 0.882·26-s − 0.192·27-s − 0.248·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 + 0.826T + 2T^{2} \) |
| 5 | \( 1 + 3.86T + 5T^{2} \) |
| 11 | \( 1 + 5.36T + 11T^{2} \) |
| 13 | \( 1 + 5.44T + 13T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 19 | \( 1 + 0.606T + 19T^{2} \) |
| 23 | \( 1 - 3.70T + 23T^{2} \) |
| 29 | \( 1 - 0.824T + 29T^{2} \) |
| 31 | \( 1 + 7.82T + 31T^{2} \) |
| 37 | \( 1 + 2.79T + 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 - 3.00T + 43T^{2} \) |
| 47 | \( 1 - 8.46T + 47T^{2} \) |
| 53 | \( 1 - 5.58T + 53T^{2} \) |
| 59 | \( 1 - 3.53T + 59T^{2} \) |
| 61 | \( 1 - 3.78T + 61T^{2} \) |
| 67 | \( 1 + 8.42T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 4.36T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 0.832T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 + 2.29T + 97T^{2} \) |
| 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
−7.949557696661515532363407690501, −7.40718790767093995782331635707, −7.24367788828178672528061064671, −5.49425082903545000288560863549, −5.05955390313838475745856934784, −4.40423811132288541725071855095, −3.57588363724950572607740980518, −2.45171524629687443072559070493, −0.800781904218305698635897359093, 0,
0.800781904218305698635897359093, 2.45171524629687443072559070493, 3.57588363724950572607740980518, 4.40423811132288541725071855095, 5.05955390313838475745856934784, 5.49425082903545000288560863549, 7.24367788828178672528061064671, 7.40718790767093995782331635707, 7.949557696661515532363407690501
