| L(s) = 1 | − 1.78·2-s + 0.0742·3-s + 1.19·4-s + 5-s − 0.132·6-s − 7-s + 1.43·8-s − 2.99·9-s − 1.78·10-s + 4.34·11-s + 0.0887·12-s − 5.31·13-s + 1.78·14-s + 0.0742·15-s − 4.96·16-s − 2.25·17-s + 5.35·18-s + 2.09·19-s + 1.19·20-s − 0.0742·21-s − 7.76·22-s + 23-s + 0.106·24-s + 25-s + 9.49·26-s − 0.444·27-s − 1.19·28-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.0428·3-s + 0.597·4-s + 0.447·5-s − 0.0541·6-s − 0.377·7-s + 0.508·8-s − 0.998·9-s − 0.565·10-s + 1.30·11-s + 0.0256·12-s − 1.47·13-s + 0.477·14-s + 0.0191·15-s − 1.24·16-s − 0.547·17-s + 1.26·18-s + 0.480·19-s + 0.267·20-s − 0.0161·21-s − 1.65·22-s + 0.208·23-s + 0.0217·24-s + 0.200·25-s + 1.86·26-s − 0.0856·27-s − 0.225·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.78T + 2T^{2} \) |
| 3 | \( 1 - 0.0742T + 3T^{2} \) |
| 11 | \( 1 - 4.34T + 11T^{2} \) |
| 13 | \( 1 + 5.31T + 13T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 19 | \( 1 - 2.09T + 19T^{2} \) |
| 29 | \( 1 + 8.32T + 29T^{2} \) |
| 31 | \( 1 - 3.81T + 31T^{2} \) |
| 37 | \( 1 + 3.34T + 37T^{2} \) |
| 41 | \( 1 - 0.389T + 41T^{2} \) |
| 43 | \( 1 + 2.40T + 43T^{2} \) |
| 47 | \( 1 - 0.385T + 47T^{2} \) |
| 53 | \( 1 + 9.38T + 53T^{2} \) |
| 59 | \( 1 + 7.25T + 59T^{2} \) |
| 61 | \( 1 - 8.11T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 3.78T + 73T^{2} \) |
| 79 | \( 1 - 5.30T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 + 18.7T + 97T^{2} \) |
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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
−9.441861027340441765284380604918, −9.295141721707443545432556282607, −8.362899255399258292531031776487, −7.35693536711150059307938131014, −6.64464165050798005104868337127, −5.53632910444451408634318049657, −4.38018957894275627252475535486, −2.90719131952393928460958889380, −1.67160727971038686107907343428, 0,
1.67160727971038686107907343428, 2.90719131952393928460958889380, 4.38018957894275627252475535486, 5.53632910444451408634318049657, 6.64464165050798005104868337127, 7.35693536711150059307938131014, 8.362899255399258292531031776487, 9.295141721707443545432556282607, 9.441861027340441765284380604918
