| L(s) = 1 | − 3·3-s − 2·4-s − 5-s − 4·7-s + 6·9-s − 11-s + 6·12-s − 6·13-s + 3·15-s + 4·16-s − 6·17-s − 4·19-s + 2·20-s + 12·21-s − 8·23-s − 4·25-s − 9·27-s + 8·28-s − 5·29-s − 10·31-s + 3·33-s + 4·35-s − 12·36-s + 18·39-s − 6·41-s − 43-s + 2·44-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 4-s − 0.447·5-s − 1.51·7-s + 2·9-s − 0.301·11-s + 1.73·12-s − 1.66·13-s + 0.774·15-s + 16-s − 1.45·17-s − 0.917·19-s + 0.447·20-s + 2.61·21-s − 1.66·23-s − 4/5·25-s − 1.73·27-s + 1.51·28-s − 0.928·29-s − 1.79·31-s + 0.522·33-s + 0.676·35-s − 2·36-s + 2.88·39-s − 0.937·41-s − 0.152·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30487 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30487 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 + T \) |
| 709 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−15.99116661188462, −15.50057154518955, −15.06089208548296, −14.31975003881175, −13.45406092245261, −13.17690368445779, −12.52472117739168, −12.34912517392855, −11.81553412919534, −11.09055760756208, −10.47193398102789, −10.12322366342164, −9.520484614008681, −9.175278232752714, −8.295848726226084, −7.456647107456234, −7.078399290624420, −6.440532200960375, −5.731025017935449, −5.519594849216842, −4.627332554525756, −4.151328449950087, −3.758079461935771, −2.591576191274265, −1.724904933357908, 0, 0, 0,
1.724904933357908, 2.591576191274265, 3.758079461935771, 4.151328449950087, 4.627332554525756, 5.519594849216842, 5.731025017935449, 6.440532200960375, 7.078399290624420, 7.456647107456234, 8.295848726226084, 9.175278232752714, 9.520484614008681, 10.12322366342164, 10.47193398102789, 11.09055760756208, 11.81553412919534, 12.34912517392855, 12.52472117739168, 13.17690368445779, 13.45406092245261, 14.31975003881175, 15.06089208548296, 15.50057154518955, 15.99116661188462
