| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 4·11-s − 2·13-s + 14-s + 16-s − 2·17-s − 4·19-s − 20-s + 4·22-s + 23-s + 25-s + 2·26-s − 28-s + 2·29-s + 8·31-s − 32-s + 2·34-s + 35-s + 6·37-s + 4·38-s + 40-s − 2·41-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 − 1.20·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 0.371·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s + 0.986·37-s + 0.648·38-s + 0.158·40-s − 0.312·41-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 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−16.26994937387013, −15.91787472739294, −15.41102552333636, −14.87951581761354, −14.32542326180466, −13.34864365477531, −13.04967518282663, −12.42579223449991, −11.78833643197386, −11.19493535561045, −10.66139039070512, −9.969894528300217, −9.755249526046559, −8.645766803503038, −8.456034833108350, −7.713925160578463, −7.177399302511104, −6.496645454583546, −5.917012323324609, −4.957821589858462, −4.464669683988915, −3.480775997304130, −2.643548486511783, −2.213054607129686, −0.8616371594983241, 0,
0.8616371594983241, 2.213054607129686, 2.643548486511783, 3.480775997304130, 4.464669683988915, 4.957821589858462, 5.917012323324609, 6.496645454583546, 7.177399302511104, 7.713925160578463, 8.456034833108350, 8.645766803503038, 9.755249526046559, 9.969894528300217, 10.66139039070512, 11.19493535561045, 11.78833643197386, 12.42579223449991, 13.04967518282663, 13.34864365477531, 14.32542326180466, 14.87951581761354, 15.41102552333636, 15.91787472739294, 16.26994937387013
