| L(s) = 1 | + 3-s − 17.2·5-s + 12.3·7-s − 26·9-s − 39.2·11-s − 18.9·13-s − 17.2·15-s + 100.·17-s − 19·19-s + 12.3·21-s − 158.·23-s + 173.·25-s − 53·27-s − 141.·29-s − 215.·31-s − 39.2·33-s − 213.·35-s − 209.·37-s − 18.9·39-s + 276.·41-s + 166.·43-s + 449.·45-s + 60.5·47-s − 190.·49-s + 100.·51-s + 76.2·53-s + 679.·55-s + ⋯ |
| L(s) = 1 | + 0.192·3-s − 1.54·5-s + 0.667·7-s − 0.962·9-s − 1.07·11-s − 0.403·13-s − 0.297·15-s + 1.42·17-s − 0.229·19-s + 0.128·21-s − 1.43·23-s + 1.39·25-s − 0.377·27-s − 0.904·29-s − 1.24·31-s − 0.207·33-s − 1.03·35-s − 0.932·37-s − 0.0777·39-s + 1.05·41-s + 0.591·43-s + 1.48·45-s + 0.187·47-s − 0.554·49-s + 0.274·51-s + 0.197·53-s + 1.66·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7898566523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7898566523\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - T + 27T^{2} \) |
| 5 | \( 1 + 17.2T + 125T^{2} \) |
| 7 | \( 1 - 12.3T + 343T^{2} \) |
| 11 | \( 1 + 39.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 18.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 100.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 141.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 215.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 209.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 276.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 166.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 60.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 76.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 575.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 91.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 444.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 943.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 569.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 336.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 603.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.63e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 319.T + 9.12e5T^{2} \) |
| 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
−9.197398166816076848853812137461, −8.204808452851937649675489846950, −7.86015310129791423137485735319, −7.31264805971668257374168860701, −5.78175010426549687860910451416, −5.13564065508851172415181808178, −4.00671011750871063449759003133, −3.28195956610430084038990752289, −2.13161113561608785090475217708, −0.42794692993666081577960750222,
0.42794692993666081577960750222, 2.13161113561608785090475217708, 3.28195956610430084038990752289, 4.00671011750871063449759003133, 5.13564065508851172415181808178, 5.78175010426549687860910451416, 7.31264805971668257374168860701, 7.86015310129791423137485735319, 8.204808452851937649675489846950, 9.197398166816076848853812137461
