| L(s) = 1 | + 0.801·2-s − 2.24·3-s − 1.35·4-s + 0.246·5-s − 1.80·6-s − 2.35·7-s − 2.69·8-s + 2.04·9-s + 0.198·10-s − 4.24·11-s + 3.04·12-s − 1.89·14-s − 0.554·15-s + 0.554·16-s + 2.15·17-s + 1.64·18-s − 0.0881·19-s − 0.335·20-s + 5.29·21-s − 3.40·22-s + 1.49·23-s + 6.04·24-s − 4.93·25-s + 2.13·27-s + 3.19·28-s + 4.63·29-s − 0.445·30-s + ⋯ |
| L(s) = 1 | + 0.567·2-s − 1.29·3-s − 0.678·4-s + 0.110·5-s − 0.735·6-s − 0.890·7-s − 0.951·8-s + 0.682·9-s + 0.0626·10-s − 1.28·11-s + 0.880·12-s − 0.505·14-s − 0.143·15-s + 0.138·16-s + 0.523·17-s + 0.387·18-s − 0.0202·19-s − 0.0749·20-s + 1.15·21-s − 0.726·22-s + 0.311·23-s + 1.23·24-s − 0.987·25-s + 0.411·27-s + 0.604·28-s + 0.859·29-s − 0.0812·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| good | 2 | \( 1 - 0.801T + 2T^{2} \) |
| 3 | \( 1 + 2.24T + 3T^{2} \) |
| 5 | \( 1 - 0.246T + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 19 | \( 1 + 0.0881T + 19T^{2} \) |
| 23 | \( 1 - 1.49T + 23T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 31 | \( 1 + 6.63T + 31T^{2} \) |
| 37 | \( 1 - 5.69T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 + 0.295T + 43T^{2} \) |
| 47 | \( 1 + 7.35T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 6.78T + 59T^{2} \) |
| 61 | \( 1 - 3.47T + 61T^{2} \) |
| 67 | \( 1 - 7.67T + 67T^{2} \) |
| 71 | \( 1 + 8.66T + 71T^{2} \) |
| 73 | \( 1 - 6.73T + 73T^{2} \) |
| 79 | \( 1 - 9.97T + 79T^{2} \) |
| 83 | \( 1 - 1.60T + 83T^{2} \) |
| 89 | \( 1 + 2.88T + 89T^{2} \) |
| 97 | \( 1 + 8.05T + 97T^{2} \) |
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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
−12.50801596828543138613328770364, −11.46927784487043750239685597641, −10.33462678075215431562594971829, −9.550834590137455943765683279258, −8.082682657034107570213469756967, −6.52528380006161753471808993267, −5.61737740930925729217237522524, −4.84691216565701289778282224049, −3.25431298166008658878705217774, 0,
3.25431298166008658878705217774, 4.84691216565701289778282224049, 5.61737740930925729217237522524, 6.52528380006161753471808993267, 8.082682657034107570213469756967, 9.550834590137455943765683279258, 10.33462678075215431562594971829, 11.46927784487043750239685597641, 12.50801596828543138613328770364
