| L(s) = 1 | + 1.30·2-s + 1.88·3-s − 0.287·4-s + 2.46·6-s − 1.28·7-s − 2.99·8-s + 0.550·9-s + 0.698·11-s − 0.541·12-s + 5.53·13-s − 1.68·14-s − 3.34·16-s − 3.43·17-s + 0.720·18-s − 3.98·19-s − 2.42·21-s + 0.914·22-s − 6.27·23-s − 5.64·24-s + 7.24·26-s − 4.61·27-s + 0.369·28-s + 6.15·29-s − 0.212·31-s + 1.61·32-s + 1.31·33-s − 4.49·34-s + ⋯ |
| L(s) = 1 | + 0.925·2-s + 1.08·3-s − 0.143·4-s + 1.00·6-s − 0.486·7-s − 1.05·8-s + 0.183·9-s + 0.210·11-s − 0.156·12-s + 1.53·13-s − 0.450·14-s − 0.835·16-s − 0.832·17-s + 0.169·18-s − 0.914·19-s − 0.529·21-s + 0.195·22-s − 1.30·23-s − 1.15·24-s + 1.42·26-s − 0.888·27-s + 0.0698·28-s + 1.14·29-s − 0.0381·31-s + 0.284·32-s + 0.229·33-s − 0.770·34-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 | \( 1 - 1.30T + 2T^{2} \) |
| 3 | \( 1 - 1.88T + 3T^{2} \) |
| 7 | \( 1 + 1.28T + 7T^{2} \) |
| 11 | \( 1 - 0.698T + 11T^{2} \) |
| 13 | \( 1 - 5.53T + 13T^{2} \) |
| 17 | \( 1 + 3.43T + 17T^{2} \) |
| 19 | \( 1 + 3.98T + 19T^{2} \) |
| 23 | \( 1 + 6.27T + 23T^{2} \) |
| 29 | \( 1 - 6.15T + 29T^{2} \) |
| 31 | \( 1 + 0.212T + 31T^{2} \) |
| 37 | \( 1 - 5.91T + 37T^{2} \) |
| 41 | \( 1 + 9.65T + 41T^{2} \) |
| 43 | \( 1 - 7.78T + 43T^{2} \) |
| 47 | \( 1 + 3.86T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 9.54T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 + 6.00T + 71T^{2} \) |
| 73 | \( 1 - 2.96T + 73T^{2} \) |
| 79 | \( 1 + 6.80T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 + 0.714T + 97T^{2} \) |
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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
−8.100195292396651014262703463581, −6.69495450362908172708769691262, −6.27035694262384584516652796072, −5.65218264162094700477302476828, −4.42984972344590286068000869069, −4.08407475926742531394183477038, −3.27970226724671181969094917213, −2.72130265748714290525847700459, −1.66314845663471014413434135036, 0,
1.66314845663471014413434135036, 2.72130265748714290525847700459, 3.27970226724671181969094917213, 4.08407475926742531394183477038, 4.42984972344590286068000869069, 5.65218264162094700477302476828, 6.27035694262384584516652796072, 6.69495450362908172708769691262, 8.100195292396651014262703463581
