| L(s) = 1 | + 2.73·3-s + 5-s + 1.26·7-s + 4.46·9-s − 1.46·11-s + 1.46·13-s + 2.73·15-s − 1.46·17-s + 4.19·19-s + 3.46·21-s + 8·23-s + 25-s + 3.99·27-s − 8.92·29-s + 2.73·31-s − 4·33-s + 1.26·35-s + 37-s + 4·39-s − 2·41-s + 6.92·43-s + 4.46·45-s + 1.26·47-s − 5.39·49-s − 4·51-s − 6·53-s − 1.46·55-s + ⋯ |
| L(s) = 1 | + 1.57·3-s + 0.447·5-s + 0.479·7-s + 1.48·9-s − 0.441·11-s + 0.406·13-s + 0.705·15-s − 0.355·17-s + 0.962·19-s + 0.755·21-s + 1.66·23-s + 0.200·25-s + 0.769·27-s − 1.65·29-s + 0.490·31-s − 0.696·33-s + 0.214·35-s + 0.164·37-s + 0.640·39-s − 0.312·41-s + 1.05·43-s + 0.665·45-s + 0.184·47-s − 0.770·49-s − 0.560·51-s − 0.824·53-s − 0.197·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.951523195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.951523195\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 + 1.46T + 17T^{2} \) |
| 19 | \( 1 - 4.19T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 8.92T + 29T^{2} \) |
| 31 | \( 1 - 2.73T + 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 - 1.26T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 0.196T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 5.26T + 79T^{2} \) |
| 83 | \( 1 - 5.26T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−8.845731847789188611225913797648, −7.997473812169786221959400788374, −7.51197447431230128722767464324, −6.70200616193571976818477463333, −5.56067969103082147221179813132, −4.80998561808021026908466254569, −3.76073345537619108141814031286, −3.01299373500729848900920590032, −2.23063275385524537251930998977, −1.26531678454265498317583031492,
1.26531678454265498317583031492, 2.23063275385524537251930998977, 3.01299373500729848900920590032, 3.76073345537619108141814031286, 4.80998561808021026908466254569, 5.56067969103082147221179813132, 6.70200616193571976818477463333, 7.51197447431230128722767464324, 7.997473812169786221959400788374, 8.845731847789188611225913797648
