| L(s) = 1 | − 2-s + 3.17·3-s + 4-s + 5-s − 3.17·6-s + 2.17·7-s − 8-s + 7.04·9-s − 10-s + 11-s + 3.17·12-s − 1.80·13-s − 2.17·14-s + 3.17·15-s + 16-s − 0.248·17-s − 7.04·18-s − 8.49·19-s + 20-s + 6.87·21-s − 22-s − 0.340·23-s − 3.17·24-s + 25-s + 1.80·26-s + 12.8·27-s + 2.17·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.83·3-s + 0.5·4-s + 0.447·5-s − 1.29·6-s + 0.820·7-s − 0.353·8-s + 2.34·9-s − 0.316·10-s + 0.301·11-s + 0.915·12-s − 0.499·13-s − 0.579·14-s + 0.818·15-s + 0.250·16-s − 0.0602·17-s − 1.66·18-s − 1.94·19-s + 0.223·20-s + 1.50·21-s − 0.213·22-s − 0.0709·23-s − 0.647·24-s + 0.200·25-s + 0.353·26-s + 2.47·27-s + 0.410·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.290523344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.290523344\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 3 | \( 1 - 3.17T + 3T^{2} \) |
| 7 | \( 1 - 2.17T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 1.80T + 13T^{2} \) |
| 17 | \( 1 + 0.248T + 17T^{2} \) |
| 19 | \( 1 + 8.49T + 19T^{2} \) |
| 23 | \( 1 + 0.340T + 23T^{2} \) |
| 29 | \( 1 + 3.80T + 29T^{2} \) |
| 31 | \( 1 - 0.539T + 31T^{2} \) |
| 37 | \( 1 + 7.14T + 37T^{2} \) |
| 41 | \( 1 - 8.43T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 7.95T + 47T^{2} \) |
| 53 | \( 1 + 8.63T + 53T^{2} \) |
| 59 | \( 1 - 0.0650T + 59T^{2} \) |
| 61 | \( 1 - 3.69T + 61T^{2} \) |
| 71 | \( 1 + 2.03T + 71T^{2} \) |
| 73 | \( 1 - 9.32T + 73T^{2} \) |
| 79 | \( 1 + 0.764T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 6.26T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
| 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
−10.23548755765434870042446625413, −9.398789168850161476873664541655, −8.746950874581208378113616097803, −8.154323512601455945515010814139, −7.37148115175624474755005957261, −6.41626192413822403707624534646, −4.77754767322717620085568850653, −3.68477551723697544216315584981, −2.37312902578741401626229846008, −1.74018750098633449322438071560,
1.74018750098633449322438071560, 2.37312902578741401626229846008, 3.68477551723697544216315584981, 4.77754767322717620085568850653, 6.41626192413822403707624534646, 7.37148115175624474755005957261, 8.154323512601455945515010814139, 8.746950874581208378113616097803, 9.398789168850161476873664541655, 10.23548755765434870042446625413
