| L(s) = 1 | + 5.15·2-s + 18.6·4-s + 8.63·5-s − 1.66·7-s + 54.7·8-s + 44.5·10-s + 28.6·13-s − 8.57·14-s + 133.·16-s + 7.86·17-s − 48.0·19-s + 160.·20-s + 153.·23-s − 50.4·25-s + 147.·26-s − 30.9·28-s + 242.·29-s + 165.·31-s + 251.·32-s + 40.5·34-s − 14.3·35-s − 68.7·37-s − 247.·38-s + 473.·40-s + 307.·41-s + 52.9·43-s + 790.·46-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 2.32·4-s + 0.772·5-s − 0.0897·7-s + 2.42·8-s + 1.40·10-s + 0.610·13-s − 0.163·14-s + 2.08·16-s + 0.112·17-s − 0.579·19-s + 1.79·20-s + 1.38·23-s − 0.403·25-s + 1.11·26-s − 0.208·28-s + 1.55·29-s + 0.961·31-s + 1.39·32-s + 0.204·34-s − 0.0693·35-s − 0.305·37-s − 1.05·38-s + 1.87·40-s + 1.17·41-s + 0.187·43-s + 2.53·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.541421046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.541421046\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 5.15T + 8T^{2} \) |
| 5 | \( 1 - 8.63T + 125T^{2} \) |
| 7 | \( 1 + 1.66T + 343T^{2} \) |
| 13 | \( 1 - 28.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 7.86T + 4.91e3T^{2} \) |
| 19 | \( 1 + 48.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 153.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 242.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 68.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 307.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 52.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 54.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 250.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 379.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 698.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 278.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 154.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 951.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 2.22T + 4.93e5T^{2} \) |
| 83 | \( 1 + 759.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 739.T + 9.12e5T^{2} \) |
| 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
−9.683936775929611749602140542033, −8.591458376815764056576082065213, −7.45941004557219669255207102291, −6.39609381988744677330449936150, −6.12589947743891151738009105889, −5.05951239044436151500816054400, −4.40125783405203335679393722680, −3.27042628894040446247863773457, −2.50068521118251704243986400340, −1.32320636950909172474630138947,
1.32320636950909172474630138947, 2.50068521118251704243986400340, 3.27042628894040446247863773457, 4.40125783405203335679393722680, 5.05951239044436151500816054400, 6.12589947743891151738009105889, 6.39609381988744677330449936150, 7.45941004557219669255207102291, 8.591458376815764056576082065213, 9.683936775929611749602140542033
