| L(s) = 1 | + 2.82·5-s + 4.82·7-s + 11-s − 0.828·13-s − 2·17-s + 2·19-s − 2.82·23-s + 3.00·25-s − 7.65·29-s − 1.65·31-s + 13.6·35-s − 3.65·37-s − 11.6·41-s + 11.6·43-s − 4.48·47-s + 16.3·49-s − 1.17·53-s + 2.82·55-s + 9.65·59-s + 8.82·61-s − 2.34·65-s − 11.3·67-s + 12.4·71-s + 9.31·73-s + 4.82·77-s − 0.828·79-s − 1.65·83-s + ⋯ |
| L(s) = 1 | + 1.26·5-s + 1.82·7-s + 0.301·11-s − 0.229·13-s − 0.485·17-s + 0.458·19-s − 0.589·23-s + 0.600·25-s − 1.42·29-s − 0.297·31-s + 2.30·35-s − 0.601·37-s − 1.82·41-s + 1.77·43-s − 0.654·47-s + 2.33·49-s − 0.160·53-s + 0.381·55-s + 1.25·59-s + 1.13·61-s − 0.290·65-s − 1.38·67-s + 1.48·71-s + 1.09·73-s + 0.550·77-s − 0.0932·79-s − 0.181·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.264510521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.264510521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 1.65T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 + 1.17T + 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 + 0.828T + 79T^{2} \) |
| 83 | \( 1 + 1.65T + 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−10.26682258990765398643964164842, −9.435783368865426291818004335827, −8.628894747738215402745051397271, −7.77074571194983908959618891358, −6.81643721365605691399258011384, −5.62040617255242187558609408904, −5.12927861593580832610504223600, −3.98309567940720693097285148112, −2.24972505342581610911783151956, −1.54354440884033386395800367605,
1.54354440884033386395800367605, 2.24972505342581610911783151956, 3.98309567940720693097285148112, 5.12927861593580832610504223600, 5.62040617255242187558609408904, 6.81643721365605691399258011384, 7.77074571194983908959618891358, 8.628894747738215402745051397271, 9.435783368865426291818004335827, 10.26682258990765398643964164842
