| L(s) = 1 | − 2-s + 4-s + 0.561·5-s − 7-s − 8-s − 0.561·10-s − 2.56·11-s + 4.56·13-s + 14-s + 16-s − 17-s + 1.12·19-s + 0.561·20-s + 2.56·22-s + 5.12·23-s − 4.68·25-s − 4.56·26-s − 28-s − 8.24·29-s + 10.2·31-s − 32-s + 34-s − 0.561·35-s − 0.561·37-s − 1.12·38-s − 0.561·40-s + 3.12·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.251·5-s − 0.377·7-s − 0.353·8-s − 0.177·10-s − 0.772·11-s + 1.26·13-s + 0.267·14-s + 0.250·16-s − 0.242·17-s + 0.257·19-s + 0.125·20-s + 0.546·22-s + 1.06·23-s − 0.936·25-s − 0.894·26-s − 0.188·28-s − 1.53·29-s + 1.84·31-s − 0.176·32-s + 0.171·34-s − 0.0949·35-s − 0.0923·37-s − 0.182·38-s − 0.0887·40-s + 0.487·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.221269639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.221269639\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 0.561T + 5T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 0.561T + 37T^{2} \) |
| 41 | \( 1 - 3.12T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 8.24T + 61T^{2} \) |
| 67 | \( 1 - 0.315T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6.80T + 73T^{2} \) |
| 79 | \( 1 + 3.68T + 79T^{2} \) |
| 83 | \( 1 + 17.9T + 83T^{2} \) |
| 89 | \( 1 - 9.68T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.959149278123731109775907740095, −8.484846715360340993838215095004, −7.59247500985697553560585836674, −6.85536474370236827451390246071, −5.99255878983429848715529982618, −5.34857207554231682230771847466, −4.05134848158383684567011442898, −3.09267287205089528279006692670, −2.08850723716282609697569315890, −0.808870634731806182246694843489,
0.808870634731806182246694843489, 2.08850723716282609697569315890, 3.09267287205089528279006692670, 4.05134848158383684567011442898, 5.34857207554231682230771847466, 5.99255878983429848715529982618, 6.85536474370236827451390246071, 7.59247500985697553560585836674, 8.484846715360340993838215095004, 8.959149278123731109775907740095
