| L(s) = 1 | − 2-s + 4-s − 5-s + 3·7-s − 8-s + 10-s + 11-s − 3·14-s + 16-s − 17-s − 8·19-s − 20-s − 22-s − 3·23-s + 25-s + 3·28-s − 3·29-s + 3·31-s − 32-s + 34-s − 3·35-s + 10·37-s + 8·38-s + 40-s − 43-s + 44-s + 3·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 0.801·14-s + 1/4·16-s − 0.242·17-s − 1.83·19-s − 0.223·20-s − 0.213·22-s − 0.625·23-s + 1/5·25-s + 0.566·28-s − 0.557·29-s + 0.538·31-s − 0.176·32-s + 0.171·34-s − 0.507·35-s + 1.64·37-s + 1.29·38-s + 0.158·40-s − 0.152·43-s + 0.150·44-s + 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.21386006726581, −15.66005832367697, −15.02268194978313, −14.64080458976736, −14.25447463362214, −13.27929096120459, −12.82832606077575, −12.09430816778778, −11.59409998109451, −10.94865996477720, −10.84724230255871, −9.862597933763781, −9.428306565451559, −8.547013180729491, −8.190157153596190, −7.875973486012057, −6.949157308157672, −6.469567022584722, −5.760452063679026, −4.858642658883683, −4.290491980501828, −3.668265419550367, −2.501726040536834, −1.973368487368723, −1.078652642192738, 0,
1.078652642192738, 1.973368487368723, 2.501726040536834, 3.668265419550367, 4.290491980501828, 4.858642658883683, 5.760452063679026, 6.469567022584722, 6.949157308157672, 7.875973486012057, 8.190157153596190, 8.547013180729491, 9.428306565451559, 9.862597933763781, 10.84724230255871, 10.94865996477720, 11.59409998109451, 12.09430816778778, 12.82832606077575, 13.27929096120459, 14.25447463362214, 14.64080458976736, 15.02268194978313, 15.66005832367697, 16.21386006726581
