| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 4·7-s − 8-s + 9-s + 10-s − 4·11-s − 12-s + 6·13-s + 4·14-s + 15-s + 16-s − 18-s − 19-s − 20-s + 4·21-s + 4·22-s + 8·23-s + 24-s + 25-s − 6·26-s − 27-s − 4·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 1.66·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.872·21-s + 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.33977957884651, −12.91163903326733, −12.75750267416130, −11.87465448014691, −11.48898641758604, −11.11139043006183, −10.49187140951610, −10.27424177194493, −9.790092553632275, −9.036246014569598, −8.842944318755701, −8.224722992109184, −7.667419609854963, −7.165928505217841, −6.631756244531942, −6.275879807927442, −5.760712806606534, −5.239591518939129, −4.547751701784447, −3.772220878268540, −3.389522762629811, −2.804931198708773, −2.240692114959082, −1.145545411147562, −0.7299599531036496, 0,
0.7299599531036496, 1.145545411147562, 2.240692114959082, 2.804931198708773, 3.389522762629811, 3.772220878268540, 4.547751701784447, 5.239591518939129, 5.760712806606534, 6.275879807927442, 6.631756244531942, 7.165928505217841, 7.667419609854963, 8.224722992109184, 8.842944318755701, 9.036246014569598, 9.790092553632275, 10.27424177194493, 10.49187140951610, 11.11139043006183, 11.48898641758604, 11.87465448014691, 12.75750267416130, 12.91163903326733, 13.33977957884651
