| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 2·11-s + 14-s + 16-s − 2·17-s + 20-s − 2·22-s − 23-s + 25-s − 28-s − 2·29-s − 6·31-s − 32-s + 2·34-s − 35-s + 4·37-s − 40-s − 6·41-s − 4·43-s + 2·44-s + 46-s + 12·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.603·11-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.223·20-s − 0.426·22-s − 0.208·23-s + 1/5·25-s − 0.188·28-s − 0.371·29-s − 1.07·31-s − 0.176·32-s + 0.342·34-s − 0.169·35-s + 0.657·37-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + 0.301·44-s + 0.147·46-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−16.66544024158571, −15.85671581294832, −15.39081790234019, −14.76234324626808, −14.21560209264811, −13.57211469224635, −13.00145545786363, −12.42019390084638, −11.79217612968511, −11.19144097351513, −10.65929213888980, −10.00053524789703, −9.481546702830156, −9.005234756967230, −8.434079739343494, −7.705948980991870, −6.935728782017501, −6.612538496304862, −5.792643152038718, −5.278859666238062, −4.213701830550370, −3.606951778692236, −2.668733369342735, −1.973822308163029, −1.125340107988271, 0,
1.125340107988271, 1.973822308163029, 2.668733369342735, 3.606951778692236, 4.213701830550370, 5.278859666238062, 5.792643152038718, 6.612538496304862, 6.935728782017501, 7.705948980991870, 8.434079739343494, 9.005234756967230, 9.481546702830156, 10.00053524789703, 10.65929213888980, 11.19144097351513, 11.79217612968511, 12.42019390084638, 13.00145545786363, 13.57211469224635, 14.21560209264811, 14.76234324626808, 15.39081790234019, 15.85671581294832, 16.66544024158571
