| L(s) = 1 | + 0.713·3-s + 2.20·5-s − 2.49·9-s + 11-s − 3.28·13-s + 1.57·15-s + 1.49·17-s + 6.91·19-s − 6.49·23-s − 0.140·25-s − 3.91·27-s − 1.64·29-s + 2.35·31-s + 0.713·33-s − 5.55·37-s − 2.34·39-s − 11.2·41-s − 5.26·43-s − 5.49·45-s − 1.49·47-s + 1.06·51-s + 0.304·53-s + 2.20·55-s + 4.93·57-s + 12.6·59-s − 12.9·61-s − 7.24·65-s + ⋯ |
| L(s) = 1 | + 0.411·3-s + 0.985·5-s − 0.830·9-s + 0.301·11-s − 0.911·13-s + 0.406·15-s + 0.361·17-s + 1.58·19-s − 1.35·23-s − 0.0281·25-s − 0.754·27-s − 0.306·29-s + 0.422·31-s + 0.124·33-s − 0.913·37-s − 0.375·39-s − 1.75·41-s − 0.803·43-s − 0.818·45-s − 0.217·47-s + 0.148·51-s + 0.0418·53-s + 0.297·55-s + 0.653·57-s + 1.64·59-s − 1.66·61-s − 0.898·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 0.713T + 3T^{2} \) |
| 5 | \( 1 - 2.20T + 5T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 17 | \( 1 - 1.49T + 17T^{2} \) |
| 19 | \( 1 - 6.91T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 + 1.64T + 29T^{2} \) |
| 31 | \( 1 - 2.35T + 31T^{2} \) |
| 37 | \( 1 + 5.55T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 5.26T + 43T^{2} \) |
| 47 | \( 1 + 1.49T + 47T^{2} \) |
| 53 | \( 1 - 0.304T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 4.57T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 8.56T + 73T^{2} \) |
| 79 | \( 1 + 4.63T + 79T^{2} \) |
| 83 | \( 1 - 1.93T + 83T^{2} \) |
| 89 | \( 1 - 3.20T + 89T^{2} \) |
| 97 | \( 1 - 1.85T + 97T^{2} \) |
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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
−7.48785519810550869431271141032, −6.74881824547214471158115257167, −5.90781834920432942799559632106, −5.46368976807143408589370787211, −4.79617934412699436339602122725, −3.64350303382618748926062735703, −3.04769882417094703490581601776, −2.19174538134469588899435708181, −1.49162625882582470249531094170, 0,
1.49162625882582470249531094170, 2.19174538134469588899435708181, 3.04769882417094703490581601776, 3.64350303382618748926062735703, 4.79617934412699436339602122725, 5.46368976807143408589370787211, 5.90781834920432942799559632106, 6.74881824547214471158115257167, 7.48785519810550869431271141032
