| L(s) = 1 | − 2-s + 2.73·3-s + 4-s + 5-s − 2.73·6-s + 7-s − 8-s + 4.46·9-s − 10-s + 11-s + 2.73·12-s − 1.46·13-s − 14-s + 2.73·15-s + 16-s + 3.46·17-s − 4.46·18-s − 2.73·19-s + 20-s + 2.73·21-s − 22-s + 1.26·23-s − 2.73·24-s + 25-s + 1.46·26-s + 3.99·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.57·3-s + 0.5·4-s + 0.447·5-s − 1.11·6-s + 0.377·7-s − 0.353·8-s + 1.48·9-s − 0.316·10-s + 0.301·11-s + 0.788·12-s − 0.406·13-s − 0.267·14-s + 0.705·15-s + 0.250·16-s + 0.840·17-s − 1.05·18-s − 0.626·19-s + 0.223·20-s + 0.596·21-s − 0.213·22-s + 0.264·23-s − 0.557·24-s + 0.200·25-s + 0.287·26-s + 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.148268086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.148268086\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 - 3.26T + 37T^{2} \) |
| 41 | \( 1 + 4.73T + 41T^{2} \) |
| 43 | \( 1 + 4.92T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 4.73T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 6.73T + 79T^{2} \) |
| 83 | \( 1 + 4.39T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 5.80T + 97T^{2} \) |
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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
−9.899320691809319646769774847936, −9.466594008553159490883476157454, −8.565951131390609539797250960296, −8.017314605183685910913203521080, −7.21720631461196602648053373698, −6.16395090877294599689990705845, −4.76045845411988619449875814218, −3.47727987592350580270132706023, −2.51578380682181850234185509181, −1.51226475364692412212879267227,
1.51226475364692412212879267227, 2.51578380682181850234185509181, 3.47727987592350580270132706023, 4.76045845411988619449875814218, 6.16395090877294599689990705845, 7.21720631461196602648053373698, 8.017314605183685910913203521080, 8.565951131390609539797250960296, 9.466594008553159490883476157454, 9.899320691809319646769774847936
