| L(s) = 1 | + 1.41·2-s + 0.517·3-s + 0.732·6-s − 2.44·7-s − 2.82·8-s − 2.73·9-s − 4.73·11-s − 2.44·13-s − 3.46·14-s − 4.00·16-s + 2.96·17-s − 3.86·18-s + 3.19·19-s − 1.26·21-s − 6.69·22-s − 1.41·23-s − 1.46·24-s − 3.46·26-s − 2.96·27-s + 2.19·29-s + 31-s − 2.44·33-s + 4.19·34-s − 0.896·37-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.298·3-s + 0.298·6-s − 0.925·7-s − 0.999·8-s − 0.910·9-s − 1.42·11-s − 0.679·13-s − 0.925·14-s − 1.00·16-s + 0.719·17-s − 0.910·18-s + 0.733·19-s − 0.276·21-s − 1.42·22-s − 0.294·23-s − 0.298·24-s − 0.679·26-s − 0.571·27-s + 0.407·29-s + 0.179·31-s − 0.426·33-s + 0.719·34-s − 0.147·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 3 | \( 1 - 0.517T + 3T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 - 2.96T + 17T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 2.19T + 29T^{2} \) |
| 37 | \( 1 + 0.896T + 37T^{2} \) |
| 41 | \( 1 + 5.53T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 - 8.76T + 47T^{2} \) |
| 53 | \( 1 - 1.27T + 53T^{2} \) |
| 59 | \( 1 + 9.92T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 2.44T + 67T^{2} \) |
| 71 | \( 1 + 7.73T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 + 4.19T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−9.834883541381461608623293399895, −9.119301394965771544124690336825, −8.125565461847610592246747914130, −7.24066680870097004245529983982, −5.90811690278825290464974306217, −5.49263062509956758706354947285, −4.41314923682166482223404288345, −3.10694841370739247092999295528, −2.75842571477002912168721833635, 0,
2.75842571477002912168721833635, 3.10694841370739247092999295528, 4.41314923682166482223404288345, 5.49263062509956758706354947285, 5.90811690278825290464974306217, 7.24066680870097004245529983982, 8.125565461847610592246747914130, 9.119301394965771544124690336825, 9.834883541381461608623293399895
