L(s) = 1 | + 2.73·2-s − 2.66·3-s + 5.50·4-s − 7.31·6-s − 4.66·7-s + 9.59·8-s + 4.12·9-s + 0.0928·11-s − 14.6·12-s − 0.640·13-s − 12.7·14-s + 15.2·16-s + 6.38·17-s + 11.3·18-s − 0.691·19-s + 12.4·21-s + 0.254·22-s − 8.33·23-s − 25.6·24-s − 1.75·26-s − 3.01·27-s − 25.6·28-s − 6.40·29-s − 0.378·31-s + 22.6·32-s − 0.247·33-s + 17.4·34-s + ⋯ |
L(s) = 1 | + 1.93·2-s − 1.54·3-s + 2.75·4-s − 2.98·6-s − 1.76·7-s + 3.39·8-s + 1.37·9-s + 0.0279·11-s − 4.24·12-s − 0.177·13-s − 3.41·14-s + 3.81·16-s + 1.54·17-s + 2.66·18-s − 0.158·19-s + 2.71·21-s + 0.0542·22-s − 1.73·23-s − 5.22·24-s − 0.343·26-s − 0.579·27-s − 4.84·28-s − 1.18·29-s − 0.0680·31-s + 4.00·32-s − 0.0431·33-s + 2.99·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 3 | \( 1 + 2.66T + 3T^{2} \) |
| 7 | \( 1 + 4.66T + 7T^{2} \) |
| 11 | \( 1 - 0.0928T + 11T^{2} \) |
| 13 | \( 1 + 0.640T + 13T^{2} \) |
| 17 | \( 1 - 6.38T + 17T^{2} \) |
| 19 | \( 1 + 0.691T + 19T^{2} \) |
| 23 | \( 1 + 8.33T + 23T^{2} \) |
| 29 | \( 1 + 6.40T + 29T^{2} \) |
| 31 | \( 1 + 0.378T + 31T^{2} \) |
| 37 | \( 1 + 4.25T + 37T^{2} \) |
| 41 | \( 1 - 5.84T + 41T^{2} \) |
| 43 | \( 1 - 1.37T + 43T^{2} \) |
| 47 | \( 1 + 9.11T + 47T^{2} \) |
| 53 | \( 1 + 6.99T + 53T^{2} \) |
| 59 | \( 1 + 7.17T + 59T^{2} \) |
| 61 | \( 1 + 0.646T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 3.38T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 0.175T + 83T^{2} \) |
| 89 | \( 1 + 8.11T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−7.20877942064616002845948533785, −6.53688810103620904502708842533, −6.01271199174249349048662696849, −5.70486771158264999849356756493, −5.01441346464503354655376415250, −4.05298714380557771181816991872, −3.55484933924901058620959258813, −2.73469117142402607538870525612, −1.51404560087753830428243428520, 0,
1.51404560087753830428243428520, 2.73469117142402607538870525612, 3.55484933924901058620959258813, 4.05298714380557771181816991872, 5.01441346464503354655376415250, 5.70486771158264999849356756493, 6.01271199174249349048662696849, 6.53688810103620904502708842533, 7.20877942064616002845948533785