| L(s) = 1 | + 1.61·2-s + 0.618·4-s − 1.61·5-s − 3·7-s − 2.23·8-s − 2.61·10-s − 0.236·11-s − 5.85·13-s − 4.85·14-s − 4.85·16-s + 2.23·17-s − 1.76·19-s − 1.00·20-s − 0.381·22-s + 7.09·23-s − 2.38·25-s − 9.47·26-s − 1.85·28-s + 1.85·29-s − 2·31-s − 3.38·32-s + 3.61·34-s + 4.85·35-s + 11.5·37-s − 2.85·38-s + 3.61·40-s − 6.38·41-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.309·4-s − 0.723·5-s − 1.13·7-s − 0.790·8-s − 0.827·10-s − 0.0711·11-s − 1.62·13-s − 1.29·14-s − 1.21·16-s + 0.542·17-s − 0.404·19-s − 0.223·20-s − 0.0814·22-s + 1.47·23-s − 0.476·25-s − 1.85·26-s − 0.350·28-s + 0.344·29-s − 0.359·31-s − 0.597·32-s + 0.620·34-s + 0.820·35-s + 1.90·37-s − 0.462·38-s + 0.572·40-s − 0.996·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 + 5.85T + 13T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 - 7.09T + 23T^{2} \) |
| 29 | \( 1 - 1.85T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 + 6.38T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 5.32T + 47T^{2} \) |
| 53 | \( 1 + 4.09T + 53T^{2} \) |
| 59 | \( 1 - 6.70T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 + 2.90T + 67T^{2} \) |
| 73 | \( 1 + 2.76T + 73T^{2} \) |
| 79 | \( 1 + 6.09T + 79T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−10.01433945052173829101299790171, −9.477598728334946657290459093244, −8.321530003034194659654683543250, −7.20292277793836895534928404363, −6.47236788248467860521252368150, −5.32901855316462828459140486812, −4.54449965267009450713520485621, −3.48004723867283049595608241912, −2.72720031897252372055370070745, 0,
2.72720031897252372055370070745, 3.48004723867283049595608241912, 4.54449965267009450713520485621, 5.32901855316462828459140486812, 6.47236788248467860521252368150, 7.20292277793836895534928404363, 8.321530003034194659654683543250, 9.477598728334946657290459093244, 10.01433945052173829101299790171
