| L(s) = 1 | − 1.51·5-s + 3.24·7-s − 3.28·11-s + 5.24·13-s − 3·17-s + 19-s + 1.51·23-s − 2.68·25-s − 1.48·29-s − 4.24·31-s − 4.93·35-s + 2.80·37-s − 10.9·41-s + 0.559·43-s − 0.0399·47-s + 3.55·49-s − 10.7·53-s + 4.99·55-s − 1.68·59-s + 5.24·61-s − 7.97·65-s − 4.68·67-s + 15.7·71-s − 12.6·73-s − 10.6·77-s + 2.80·79-s + 3·83-s + ⋯ |
| L(s) = 1 | − 0.679·5-s + 1.22·7-s − 0.991·11-s + 1.45·13-s − 0.727·17-s + 0.229·19-s + 0.316·23-s − 0.537·25-s − 0.274·29-s − 0.763·31-s − 0.834·35-s + 0.461·37-s − 1.71·41-s + 0.0853·43-s − 0.00582·47-s + 0.508·49-s − 1.47·53-s + 0.674·55-s − 0.219·59-s + 0.672·61-s − 0.989·65-s − 0.572·67-s + 1.86·71-s − 1.47·73-s − 1.21·77-s + 0.316·79-s + 0.329·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 1.51T + 5T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 11 | \( 1 + 3.28T + 11T^{2} \) |
| 13 | \( 1 - 5.24T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 23 | \( 1 - 1.51T + 23T^{2} \) |
| 29 | \( 1 + 1.48T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 - 2.80T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 0.559T + 43T^{2} \) |
| 47 | \( 1 + 0.0399T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 1.68T + 59T^{2} \) |
| 61 | \( 1 - 5.24T + 61T^{2} \) |
| 67 | \( 1 + 4.68T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 2.80T + 79T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 0.310T + 97T^{2} \) |
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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
−7.65411773068246985387021836758, −6.88750351677078673562375945626, −6.03022518536269191809006266105, −5.27902136311417534775923736830, −4.68375946962117402394146653693, −3.91036244353392625556832780410, −3.21440992928650902072895042698, −2.10583955856185302424786242645, −1.32044708444679753634548519576, 0,
1.32044708444679753634548519576, 2.10583955856185302424786242645, 3.21440992928650902072895042698, 3.91036244353392625556832780410, 4.68375946962117402394146653693, 5.27902136311417534775923736830, 6.03022518536269191809006266105, 6.88750351677078673562375945626, 7.65411773068246985387021836758
