| L(s) = 1 | + 5-s − 0.267·7-s − 3.46·11-s + 0.267·13-s − 3.46·17-s − 5.92·19-s + 6.46·23-s + 25-s + 6.92·29-s + 1.46·31-s − 0.267·35-s + 8·37-s − 5.19·41-s − 5.46·43-s − 0.464·47-s − 6.92·49-s − 5.53·53-s − 3.46·55-s + 8.66·59-s + 12.3·61-s + 0.267·65-s − 8·67-s + 6·71-s − 14.3·73-s + 0.928·77-s + 14.3·79-s + 15.4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.101·7-s − 1.04·11-s + 0.0743·13-s − 0.840·17-s − 1.36·19-s + 1.34·23-s + 0.200·25-s + 1.28·29-s + 0.262·31-s − 0.0452·35-s + 1.31·37-s − 0.811·41-s − 0.833·43-s − 0.0676·47-s − 0.989·49-s − 0.760·53-s − 0.467·55-s + 1.12·59-s + 1.58·61-s + 0.0332·65-s − 0.977·67-s + 0.712·71-s − 1.68·73-s + 0.105·77-s + 1.61·79-s + 1.69·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.735826742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.735826742\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 0.267T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 0.267T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 5.92T + 19T^{2} \) |
| 23 | \( 1 - 6.46T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 5.19T + 41T^{2} \) |
| 43 | \( 1 + 5.46T + 43T^{2} \) |
| 47 | \( 1 + 0.464T + 47T^{2} \) |
| 53 | \( 1 + 5.53T + 53T^{2} \) |
| 59 | \( 1 - 8.66T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−8.197285997615300830375974048381, −7.22699076806315844693263866293, −6.51468662645145753252418346731, −6.07082437892846404404910559051, −4.84663124938125165389894799694, −4.78929914382997569730204983240, −3.49988654084750094469624131866, −2.65193786923731171444655459082, −1.99381315136792885981583950205, −0.66188319967947064809871692706,
0.66188319967947064809871692706, 1.99381315136792885981583950205, 2.65193786923731171444655459082, 3.49988654084750094469624131866, 4.78929914382997569730204983240, 4.84663124938125165389894799694, 6.07082437892846404404910559051, 6.51468662645145753252418346731, 7.22699076806315844693263866293, 8.197285997615300830375974048381
