| L(s) = 1 | − 2-s + 0.381·3-s + 4-s − 2.23·5-s − 0.381·6-s + 2·7-s − 8-s − 2.85·9-s + 2.23·10-s − 3.38·11-s + 0.381·12-s + 5·13-s − 2·14-s − 0.854·15-s + 16-s + 2.23·17-s + 2.85·18-s + 19-s − 2.23·20-s + 0.763·21-s + 3.38·22-s − 8.23·23-s − 0.381·24-s − 5·26-s − 2.23·27-s + 2·28-s − 4.85·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.220·3-s + 0.5·4-s − 0.999·5-s − 0.155·6-s + 0.755·7-s − 0.353·8-s − 0.951·9-s + 0.707·10-s − 1.01·11-s + 0.110·12-s + 1.38·13-s − 0.534·14-s − 0.220·15-s + 0.250·16-s + 0.542·17-s + 0.672·18-s + 0.229·19-s − 0.499·20-s + 0.166·21-s + 0.721·22-s − 1.71·23-s − 0.0779·24-s − 0.980·26-s − 0.430·27-s + 0.377·28-s − 0.901·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8197318790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8197318790\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 0.381T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 + 4.85T + 29T^{2} \) |
| 31 | \( 1 + 9.56T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 + 1.47T + 41T^{2} \) |
| 43 | \( 1 + 6.85T + 43T^{2} \) |
| 47 | \( 1 - 8.23T + 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 - 5.61T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 4.09T + 83T^{2} \) |
| 89 | \( 1 - 8.23T + 89T^{2} \) |
| 97 | \( 1 - 13T + 97T^{2} \) |
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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
−8.008632027320983827372237814065, −7.59718875760134350482740927557, −6.56250965467665071339209839976, −5.63571214266717923875113202153, −5.32461197945247567935261549480, −3.89176832979567537172400534966, −3.66781147261883875441922424223, −2.54010716576137028717504052442, −1.73124805448654816609977502415, −0.48337474921107922635807982564,
0.48337474921107922635807982564, 1.73124805448654816609977502415, 2.54010716576137028717504052442, 3.66781147261883875441922424223, 3.89176832979567537172400534966, 5.32461197945247567935261549480, 5.63571214266717923875113202153, 6.56250965467665071339209839976, 7.59718875760134350482740927557, 8.008632027320983827372237814065
