| L(s) = 1 | + 1.70·2-s − 0.930·3-s + 0.917·4-s − 1.58·6-s − 1.50·7-s − 1.84·8-s − 2.13·9-s + 1.68·11-s − 0.854·12-s − 4.04·13-s − 2.57·14-s − 4.99·16-s + 0.146·17-s − 3.64·18-s − 3.59·19-s + 1.40·21-s + 2.88·22-s − 4.77·23-s + 1.72·24-s − 6.91·26-s + 4.77·27-s − 1.38·28-s + 6.52·29-s − 31-s − 4.83·32-s − 1.57·33-s + 0.250·34-s + ⋯ |
| L(s) = 1 | + 1.20·2-s − 0.537·3-s + 0.458·4-s − 0.649·6-s − 0.569·7-s − 0.653·8-s − 0.711·9-s + 0.509·11-s − 0.246·12-s − 1.12·13-s − 0.688·14-s − 1.24·16-s + 0.0355·17-s − 0.859·18-s − 0.823·19-s + 0.306·21-s + 0.615·22-s − 0.994·23-s + 0.351·24-s − 1.35·26-s + 0.919·27-s − 0.261·28-s + 1.21·29-s − 0.179·31-s − 0.854·32-s − 0.273·33-s + 0.0429·34-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 | \( 1 - 1.70T + 2T^{2} \) |
| 3 | \( 1 + 0.930T + 3T^{2} \) |
| 7 | \( 1 + 1.50T + 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 - 0.146T + 17T^{2} \) |
| 19 | \( 1 + 3.59T + 19T^{2} \) |
| 23 | \( 1 + 4.77T + 23T^{2} \) |
| 29 | \( 1 - 6.52T + 29T^{2} \) |
| 37 | \( 1 + 5.06T + 37T^{2} \) |
| 41 | \( 1 - 0.252T + 41T^{2} \) |
| 43 | \( 1 + 0.0635T + 43T^{2} \) |
| 47 | \( 1 - 0.392T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 7.08T + 59T^{2} \) |
| 61 | \( 1 + 0.825T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 8.86T + 73T^{2} \) |
| 79 | \( 1 - 4.74T + 79T^{2} \) |
| 83 | \( 1 - 4.20T + 83T^{2} \) |
| 89 | \( 1 + 7.58T + 89T^{2} \) |
| 97 | \( 1 - 2.57T + 97T^{2} \) |
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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
−9.975922939496600173218152360422, −9.113816236610605692160659979078, −8.147534453589045635833990574484, −6.73071593561153289638563876732, −6.24342223938293034258541391506, −5.30341750389454706753076422634, −4.53053395050048119457076112782, −3.47644305980033908317040639904, −2.43759836091408554312243442144, 0,
2.43759836091408554312243442144, 3.47644305980033908317040639904, 4.53053395050048119457076112782, 5.30341750389454706753076422634, 6.24342223938293034258541391506, 6.73071593561153289638563876732, 8.147534453589045635833990574484, 9.113816236610605692160659979078, 9.975922939496600173218152360422
