| L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s − 7-s + 3·8-s + 9-s − 10-s − 12-s − 2·13-s + 14-s + 15-s − 16-s − 18-s − 20-s − 21-s − 4·23-s + 3·24-s + 25-s + 2·26-s + 27-s + 28-s + 6·29-s − 30-s − 5·32-s − 35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.235·18-s − 0.223·20-s − 0.218·21-s − 0.834·23-s + 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.182·30-s − 0.883·32-s − 0.169·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.38179299224084, −14.82559069533775, −14.24202506390808, −13.72176793219196, −13.46658162121180, −12.86379601592065, −12.14735473095220, −11.84095779216562, −10.68834135295585, −10.45674896242216, −9.808922731134307, −9.528789591130948, −8.892311709783544, −8.411084175861303, −7.885736816833070, −7.322622569536564, −6.634943713226651, −6.073668740103516, −5.120917119224832, −4.759585242895457, −3.942000193392801, −3.330216591184412, −2.474118946932945, −1.828874148346721, −0.9734027505019840, 0,
0.9734027505019840, 1.828874148346721, 2.474118946932945, 3.330216591184412, 3.942000193392801, 4.759585242895457, 5.120917119224832, 6.073668740103516, 6.634943713226651, 7.322622569536564, 7.885736816833070, 8.411084175861303, 8.892311709783544, 9.528789591130948, 9.808922731134307, 10.45674896242216, 10.68834135295585, 11.84095779216562, 12.14735473095220, 12.86379601592065, 13.46658162121180, 13.72176793219196, 14.24202506390808, 14.82559069533775, 15.38179299224084
