| L(s) = 1 | + 3-s + 0.414·5-s + 7-s + 9-s − 11-s − 1.58·13-s + 0.414·15-s − 17-s + 3.58·19-s + 21-s + 3.82·23-s − 4.82·25-s + 27-s + 6.82·29-s + 9.89·31-s − 33-s + 0.414·35-s − 8.24·37-s − 1.58·39-s − 4.07·41-s + 0.171·43-s + 0.414·45-s + 2.58·47-s + 49-s − 51-s − 2.24·53-s − 0.414·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.185·5-s + 0.377·7-s + 0.333·9-s − 0.301·11-s − 0.439·13-s + 0.106·15-s − 0.242·17-s + 0.822·19-s + 0.218·21-s + 0.798·23-s − 0.965·25-s + 0.192·27-s + 1.26·29-s + 1.77·31-s − 0.174·33-s + 0.0700·35-s − 1.35·37-s − 0.253·39-s − 0.635·41-s + 0.0261·43-s + 0.0617·45-s + 0.377·47-s + 0.142·49-s − 0.140·51-s − 0.308·53-s − 0.0558·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.733561018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.733561018\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 0.414T + 5T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 23 | \( 1 - 3.82T + 23T^{2} \) |
| 29 | \( 1 - 6.82T + 29T^{2} \) |
| 31 | \( 1 - 9.89T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 4.07T + 41T^{2} \) |
| 43 | \( 1 - 0.171T + 43T^{2} \) |
| 47 | \( 1 - 2.58T + 47T^{2} \) |
| 53 | \( 1 + 2.24T + 53T^{2} \) |
| 59 | \( 1 - 3.75T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 7.65T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 3.17T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 1.65T + 97T^{2} \) |
| 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
−8.347800100291707938316455271083, −7.38449536879295335366965804200, −6.89420283550869055549455107642, −5.98437824275424349483824455126, −5.07670968508387231612777342469, −4.58951721246886266343158634929, −3.54017576023724474879045079164, −2.77937453064625149295467605284, −1.98438468724894009061767019571, −0.870825852437118525615580797732,
0.870825852437118525615580797732, 1.98438468724894009061767019571, 2.77937453064625149295467605284, 3.54017576023724474879045079164, 4.58951721246886266343158634929, 5.07670968508387231612777342469, 5.98437824275424349483824455126, 6.89420283550869055549455107642, 7.38449536879295335366965804200, 8.347800100291707938316455271083
