| L(s) = 1 | − 1.59·2-s + 3-s + 0.545·4-s − 5-s − 1.59·6-s − 0.379·7-s + 2.32·8-s + 9-s + 1.59·10-s − 3.21·11-s + 0.545·12-s + 2.18·13-s + 0.605·14-s − 15-s − 4.79·16-s − 4.28·17-s − 1.59·18-s − 3.09·19-s − 0.545·20-s − 0.379·21-s + 5.13·22-s + 2.77·23-s + 2.32·24-s + 25-s − 3.47·26-s + 27-s − 0.207·28-s + ⋯ |
| L(s) = 1 | − 1.12·2-s + 0.577·3-s + 0.272·4-s − 0.447·5-s − 0.651·6-s − 0.143·7-s + 0.820·8-s + 0.333·9-s + 0.504·10-s − 0.970·11-s + 0.157·12-s + 0.604·13-s + 0.161·14-s − 0.258·15-s − 1.19·16-s − 1.04·17-s − 0.376·18-s − 0.709·19-s − 0.122·20-s − 0.0828·21-s + 1.09·22-s + 0.579·23-s + 0.473·24-s + 0.200·25-s − 0.682·26-s + 0.192·27-s − 0.0391·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 1.59T + 2T^{2} \) |
| 7 | \( 1 + 0.379T + 7T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 13 | \( 1 - 2.18T + 13T^{2} \) |
| 17 | \( 1 + 4.28T + 17T^{2} \) |
| 19 | \( 1 + 3.09T + 19T^{2} \) |
| 23 | \( 1 - 2.77T + 23T^{2} \) |
| 29 | \( 1 - 8.96T + 29T^{2} \) |
| 31 | \( 1 + 9.54T + 31T^{2} \) |
| 37 | \( 1 - 8.61T + 37T^{2} \) |
| 41 | \( 1 - 7.49T + 41T^{2} \) |
| 43 | \( 1 - 6.86T + 43T^{2} \) |
| 47 | \( 1 - 5.18T + 47T^{2} \) |
| 53 | \( 1 + 1.55T + 53T^{2} \) |
| 59 | \( 1 - 2.60T + 59T^{2} \) |
| 61 | \( 1 + 2.94T + 61T^{2} \) |
| 67 | \( 1 + 2.09T + 67T^{2} \) |
| 71 | \( 1 + 9.97T + 71T^{2} \) |
| 73 | \( 1 + 3.02T + 73T^{2} \) |
| 79 | \( 1 - 1.33T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 + 8.93T + 97T^{2} \) |
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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
−7.912109879845822937149006124004, −7.32823930077381052245412503923, −6.63353377763545886465146905789, −5.67351704008925386756948423276, −4.52902002900773312967934965615, −4.18932668518233635694372713491, −2.96540676640178391673889565548, −2.24994450138709963805795198672, −1.11188020181203075567286742183, 0,
1.11188020181203075567286742183, 2.24994450138709963805795198672, 2.96540676640178391673889565548, 4.18932668518233635694372713491, 4.52902002900773312967934965615, 5.67351704008925386756948423276, 6.63353377763545886465146905789, 7.32823930077381052245412503923, 7.912109879845822937149006124004
