| L(s) = 1 | + 0.0774·2-s + 3-s − 1.99·4-s − 5-s + 0.0774·6-s + 3.48·7-s − 0.309·8-s + 9-s − 0.0774·10-s + 0.122·11-s − 1.99·12-s − 0.150·13-s + 0.269·14-s − 15-s + 3.96·16-s + 2.68·17-s + 0.0774·18-s − 3.20·19-s + 1.99·20-s + 3.48·21-s + 0.00949·22-s − 0.309·24-s + 25-s − 0.0116·26-s + 27-s − 6.94·28-s − 0.318·29-s + ⋯ |
| L(s) = 1 | + 0.0547·2-s + 0.577·3-s − 0.997·4-s − 0.447·5-s + 0.0316·6-s + 1.31·7-s − 0.109·8-s + 0.333·9-s − 0.0244·10-s + 0.0369·11-s − 0.575·12-s − 0.0416·13-s + 0.0720·14-s − 0.258·15-s + 0.991·16-s + 0.652·17-s + 0.0182·18-s − 0.735·19-s + 0.445·20-s + 0.759·21-s + 0.00202·22-s − 0.0631·24-s + 0.200·25-s − 0.00228·26-s + 0.192·27-s − 1.31·28-s − 0.0592·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.228252149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.228252149\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.0774T + 2T^{2} \) |
| 7 | \( 1 - 3.48T + 7T^{2} \) |
| 11 | \( 1 - 0.122T + 11T^{2} \) |
| 13 | \( 1 + 0.150T + 13T^{2} \) |
| 17 | \( 1 - 2.68T + 17T^{2} \) |
| 19 | \( 1 + 3.20T + 19T^{2} \) |
| 29 | \( 1 + 0.318T + 29T^{2} \) |
| 31 | \( 1 - 4.15T + 31T^{2} \) |
| 37 | \( 1 + 2.32T + 37T^{2} \) |
| 41 | \( 1 - 0.522T + 41T^{2} \) |
| 43 | \( 1 - 3.64T + 43T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 1.19T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 7.64T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 0.151T + 83T^{2} \) |
| 89 | \( 1 + 3.87T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81770786218254176426058800548, −7.59054582073794941841277277732, −6.47672561560669279854348314067, −5.52956020175883978748697539741, −4.88622676765938366172740296071, −4.28137589462156706769383932628, −3.71539969492908394578477116308, −2.74659658607351565627492573733, −1.70635240712906758528553529979, −0.75472547024661415141438754308,
0.75472547024661415141438754308, 1.70635240712906758528553529979, 2.74659658607351565627492573733, 3.71539969492908394578477116308, 4.28137589462156706769383932628, 4.88622676765938366172740296071, 5.52956020175883978748697539741, 6.47672561560669279854348314067, 7.59054582073794941841277277732, 7.81770786218254176426058800548
