| L(s) = 1 | + 2-s + 4-s − 2.53·5-s − 2.34·7-s + 8-s − 2.53·10-s + 4.29·11-s + 0.758·13-s − 2.34·14-s + 16-s − 4.92·17-s − 7.24·19-s − 2.53·20-s + 4.29·22-s + 0.305·23-s + 1.41·25-s + 0.758·26-s − 2.34·28-s + 3.10·29-s − 8.59·31-s + 32-s − 4.92·34-s + 5.94·35-s − 7.57·37-s − 7.24·38-s − 2.53·40-s − 4.90·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.13·5-s − 0.887·7-s + 0.353·8-s − 0.800·10-s + 1.29·11-s + 0.210·13-s − 0.627·14-s + 0.250·16-s − 1.19·17-s − 1.66·19-s − 0.566·20-s + 0.914·22-s + 0.0636·23-s + 0.282·25-s + 0.148·26-s − 0.443·28-s + 0.576·29-s − 1.54·31-s + 0.176·32-s − 0.843·34-s + 1.00·35-s − 1.24·37-s − 1.17·38-s − 0.400·40-s − 0.765·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.53T + 5T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 11 | \( 1 - 4.29T + 11T^{2} \) |
| 13 | \( 1 - 0.758T + 13T^{2} \) |
| 17 | \( 1 + 4.92T + 17T^{2} \) |
| 19 | \( 1 + 7.24T + 19T^{2} \) |
| 23 | \( 1 - 0.305T + 23T^{2} \) |
| 29 | \( 1 - 3.10T + 29T^{2} \) |
| 31 | \( 1 + 8.59T + 31T^{2} \) |
| 37 | \( 1 + 7.57T + 37T^{2} \) |
| 41 | \( 1 + 4.90T + 41T^{2} \) |
| 43 | \( 1 + 1.63T + 43T^{2} \) |
| 47 | \( 1 - 1.81T + 47T^{2} \) |
| 53 | \( 1 + 0.573T + 53T^{2} \) |
| 59 | \( 1 + 5.47T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 0.0418T + 67T^{2} \) |
| 71 | \( 1 - 4.20T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 7.37T + 79T^{2} \) |
| 83 | \( 1 + 6.82T + 83T^{2} \) |
| 89 | \( 1 + 7.92T + 89T^{2} \) |
| 97 | \( 1 + 3.28T + 97T^{2} \) |
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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
−8.892080039554351731487902639479, −8.456067403753439941289777231535, −7.12747836562599051746916660154, −6.72839680947481487061195827408, −5.94238703589980724413417064761, −4.59010876876488672978838468459, −3.95604132269624399904265074776, −3.31615700978255030754866748572, −1.91903985725952206449668302401, 0,
1.91903985725952206449668302401, 3.31615700978255030754866748572, 3.95604132269624399904265074776, 4.59010876876488672978838468459, 5.94238703589980724413417064761, 6.72839680947481487061195827408, 7.12747836562599051746916660154, 8.456067403753439941289777231535, 8.892080039554351731487902639479
