| L(s) = 1 | + 0.983·3-s − 5-s − 2.43·7-s − 2.03·9-s + 4.69·11-s + 1.64·13-s − 0.983·15-s + 0.351·17-s − 1.05·19-s − 2.39·21-s + 0.237·23-s + 25-s − 4.94·27-s + 10.7·29-s + 2.17·31-s + 4.61·33-s + 2.43·35-s − 7.57·37-s + 1.61·39-s − 9.89·41-s − 0.895·43-s + 2.03·45-s + 3.18·47-s − 1.07·49-s + 0.345·51-s + 7.34·53-s − 4.69·55-s + ⋯ |
| L(s) = 1 | + 0.567·3-s − 0.447·5-s − 0.920·7-s − 0.677·9-s + 1.41·11-s + 0.456·13-s − 0.253·15-s + 0.0851·17-s − 0.242·19-s − 0.522·21-s + 0.0495·23-s + 0.200·25-s − 0.952·27-s + 1.98·29-s + 0.389·31-s + 0.802·33-s + 0.411·35-s − 1.24·37-s + 0.259·39-s − 1.54·41-s − 0.136·43-s + 0.303·45-s + 0.464·47-s − 0.153·49-s + 0.0483·51-s + 1.00·53-s − 0.632·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.874987538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.874987538\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 - 0.983T + 3T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 - 4.69T + 11T^{2} \) |
| 13 | \( 1 - 1.64T + 13T^{2} \) |
| 17 | \( 1 - 0.351T + 17T^{2} \) |
| 19 | \( 1 + 1.05T + 19T^{2} \) |
| 23 | \( 1 - 0.237T + 23T^{2} \) |
| 29 | \( 1 - 10.7T + 29T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 37 | \( 1 + 7.57T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 + 0.895T + 43T^{2} \) |
| 47 | \( 1 - 3.18T + 47T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 + 9.06T + 59T^{2} \) |
| 61 | \( 1 + 2.85T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 8.27T + 73T^{2} \) |
| 79 | \( 1 - 7.87T + 79T^{2} \) |
| 83 | \( 1 + 4.21T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
| 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
−8.321848143371193182331474981730, −7.34625452700282462692547638011, −6.48011735460687993588802991344, −6.31103342097184278439758934316, −5.15883055368778061002080465908, −4.24416281161017834518734120709, −3.43347458874121699362309060673, −3.08911207392186707055751885319, −1.90961036462814447028481754232, −0.69426452487163097349480740355,
0.69426452487163097349480740355, 1.90961036462814447028481754232, 3.08911207392186707055751885319, 3.43347458874121699362309060673, 4.24416281161017834518734120709, 5.15883055368778061002080465908, 6.31103342097184278439758934316, 6.48011735460687993588802991344, 7.34625452700282462692547638011, 8.321848143371193182331474981730
