| L(s) = 1 | − 2·2-s + 2·4-s + 5-s + 7-s − 2·10-s − 3·11-s − 2·14-s − 4·16-s + 19-s + 2·20-s + 6·22-s + 6·23-s + 25-s + 2·28-s + 5·29-s + 8·32-s + 35-s + 6·37-s − 2·38-s + 41-s − 2·43-s − 6·44-s − 12·46-s + 2·47-s + 49-s − 2·50-s − 6·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s + 0.377·7-s − 0.632·10-s − 0.904·11-s − 0.534·14-s − 16-s + 0.229·19-s + 0.447·20-s + 1.27·22-s + 1.25·23-s + 1/5·25-s + 0.377·28-s + 0.928·29-s + 1.41·32-s + 0.169·35-s + 0.986·37-s − 0.324·38-s + 0.156·41-s − 0.304·43-s − 0.904·44-s − 1.76·46-s + 0.291·47-s + 1/7·49-s − 0.282·50-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.14623932743314, −13.58223274221064, −13.06298144099250, −12.72237539127511, −11.97178749629751, −11.36068580937544, −10.94294652223168, −10.63460277878159, −9.893205160360545, −9.750048156554850, −9.138796595746716, −8.439946299947103, −8.345477496250470, −7.639784433391342, −7.173384882382810, −6.712717403130070, −6.021853250362272, −5.358567079700637, −4.841029492748602, −4.351384705486973, −3.375442956493443, −2.616851819264239, −2.277374655149051, −1.333068731040959, −0.9289303362926632, 0,
0.9289303362926632, 1.333068731040959, 2.277374655149051, 2.616851819264239, 3.375442956493443, 4.351384705486973, 4.841029492748602, 5.358567079700637, 6.021853250362272, 6.712717403130070, 7.173384882382810, 7.639784433391342, 8.345477496250470, 8.439946299947103, 9.138796595746716, 9.750048156554850, 9.893205160360545, 10.63460277878159, 10.94294652223168, 11.36068580937544, 11.97178749629751, 12.72237539127511, 13.06298144099250, 13.58223274221064, 14.14623932743314
