| L(s) = 1 | − 3.92·2-s − 3·3-s + 7.43·4-s − 7.76·5-s + 11.7·6-s + 18.1·7-s + 2.23·8-s + 9·9-s + 30.4·10-s + 11·11-s − 22.2·12-s − 74.3·13-s − 71.2·14-s + 23.2·15-s − 68.2·16-s − 43.7·17-s − 35.3·18-s − 19·19-s − 57.6·20-s − 54.4·21-s − 43.2·22-s + 154.·23-s − 6.69·24-s − 64.7·25-s + 292.·26-s − 27·27-s + 134.·28-s + ⋯ |
| L(s) = 1 | − 1.38·2-s − 0.577·3-s + 0.928·4-s − 0.694·5-s + 0.801·6-s + 0.979·7-s + 0.0986·8-s + 0.333·9-s + 0.964·10-s + 0.301·11-s − 0.536·12-s − 1.58·13-s − 1.36·14-s + 0.400·15-s − 1.06·16-s − 0.623·17-s − 0.462·18-s − 0.229·19-s − 0.644·20-s − 0.565·21-s − 0.418·22-s + 1.40·23-s − 0.0569·24-s − 0.517·25-s + 2.20·26-s − 0.192·27-s + 0.909·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 3.92T + 8T^{2} \) |
| 5 | \( 1 + 7.76T + 125T^{2} \) |
| 7 | \( 1 - 18.1T + 343T^{2} \) |
| 13 | \( 1 + 74.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 43.7T + 4.91e3T^{2} \) |
| 23 | \( 1 - 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 23.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 177.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 362.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 65.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 102.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 266.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 798.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 116.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 250.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 39.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 304.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 669.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3T + 9.12e5T^{2} \) |
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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
−9.698928966328295298031823075529, −8.927569632156014826950654277347, −7.962099161754669112063721262948, −7.43582731705345550992645774001, −6.55806573196990986105427916760, −4.96264888011349662314614636360, −4.39133617228279752323850098610, −2.44500900421885966693031233678, −1.12674349783297304224821794578, 0,
1.12674349783297304224821794578, 2.44500900421885966693031233678, 4.39133617228279752323850098610, 4.96264888011349662314614636360, 6.55806573196990986105427916760, 7.43582731705345550992645774001, 7.962099161754669112063721262948, 8.927569632156014826950654277347, 9.698928966328295298031823075529
