| L(s) = 1 | − 2-s − 1.67·3-s + 4-s − 3.90·5-s + 1.67·6-s − 0.186·7-s − 8-s − 0.191·9-s + 3.90·10-s + 11-s − 1.67·12-s − 6.91·13-s + 0.186·14-s + 6.55·15-s + 16-s − 0.375·17-s + 0.191·18-s + 0.115·19-s − 3.90·20-s + 0.312·21-s − 22-s + 2.11·23-s + 1.67·24-s + 10.2·25-s + 6.91·26-s + 5.34·27-s − 0.186·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.967·3-s + 0.5·4-s − 1.74·5-s + 0.684·6-s − 0.0704·7-s − 0.353·8-s − 0.0637·9-s + 1.23·10-s + 0.301·11-s − 0.483·12-s − 1.91·13-s + 0.0497·14-s + 1.69·15-s + 0.250·16-s − 0.0911·17-s + 0.0450·18-s + 0.0265·19-s − 0.874·20-s + 0.0681·21-s − 0.213·22-s + 0.442·23-s + 0.342·24-s + 2.05·25-s + 1.35·26-s + 1.02·27-s − 0.0352·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 1.67T + 3T^{2} \) |
| 5 | \( 1 + 3.90T + 5T^{2} \) |
| 7 | \( 1 + 0.186T + 7T^{2} \) |
| 13 | \( 1 + 6.91T + 13T^{2} \) |
| 17 | \( 1 + 0.375T + 17T^{2} \) |
| 19 | \( 1 - 0.115T + 19T^{2} \) |
| 23 | \( 1 - 2.11T + 23T^{2} \) |
| 29 | \( 1 - 1.10T + 29T^{2} \) |
| 31 | \( 1 + 8.13T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 3.93T + 41T^{2} \) |
| 43 | \( 1 - 5.46T + 43T^{2} \) |
| 47 | \( 1 + 3.16T + 47T^{2} \) |
| 53 | \( 1 - 9.17T + 53T^{2} \) |
| 59 | \( 1 + 5.00T + 59T^{2} \) |
| 61 | \( 1 - 2.17T + 61T^{2} \) |
| 67 | \( 1 + 1.99T + 67T^{2} \) |
| 71 | \( 1 - 3.16T + 71T^{2} \) |
| 73 | \( 1 + 3.04T + 73T^{2} \) |
| 79 | \( 1 + 2.40T + 79T^{2} \) |
| 83 | \( 1 - 8.81T + 83T^{2} \) |
| 89 | \( 1 - 1.79T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
| 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
−7.80061970454317850725355611077, −7.40969240474444852540220077271, −6.81261137116309429125385706801, −5.89055849466343625360488905767, −4.94921788480747560857252219732, −4.38330155545341300061764102387, −3.33589032089565199226279284462, −2.43303946975017646232021021909, −0.78718281182167496016790352000, 0,
0.78718281182167496016790352000, 2.43303946975017646232021021909, 3.33589032089565199226279284462, 4.38330155545341300061764102387, 4.94921788480747560857252219732, 5.89055849466343625360488905767, 6.81261137116309429125385706801, 7.40969240474444852540220077271, 7.80061970454317850725355611077
