| L(s) = 1 | − 2.58·2-s − 1.31·4-s + 5·5-s − 22.8·7-s + 24.0·8-s − 12.9·10-s + 11.0·11-s − 11.6·13-s + 59.1·14-s − 51.7·16-s + 10.0·17-s + 117.·19-s − 6.59·20-s − 28.6·22-s + 172.·23-s + 25·25-s + 30.0·26-s + 30.1·28-s + 178.·29-s + 140.·31-s − 59.0·32-s − 26.0·34-s − 114.·35-s + 250.·37-s − 304.·38-s + 120.·40-s + 361.·41-s + ⋯ |
| L(s) = 1 | − 0.913·2-s − 0.164·4-s + 0.447·5-s − 1.23·7-s + 1.06·8-s − 0.408·10-s + 0.303·11-s − 0.248·13-s + 1.12·14-s − 0.808·16-s + 0.143·17-s + 1.42·19-s − 0.0737·20-s − 0.277·22-s + 1.56·23-s + 0.200·25-s + 0.226·26-s + 0.203·28-s + 1.14·29-s + 0.814·31-s − 0.326·32-s − 0.131·34-s − 0.552·35-s + 1.11·37-s − 1.30·38-s + 0.476·40-s + 1.37·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8701778633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8701778633\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 + 2.58T + 8T^{2} \) |
| 7 | \( 1 + 22.8T + 343T^{2} \) |
| 11 | \( 1 - 11.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 11.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 10.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 117.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 250.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 361.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 600.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 201.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 415.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 54.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 531.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 933.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 560.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 810.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 538.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 686.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 714.T + 9.12e5T^{2} \) |
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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
−12.94078456086323210845634924299, −11.60092514191548874754994905469, −10.21901100310839782682038709612, −9.640763673825080410958123124545, −8.839837742895997453616951799801, −7.49354775321479957545988309214, −6.39714317555827353831646835322, −4.86622130777510368577436397152, −3.07916926871191843506904711816, −0.928288195279676937966404069461,
0.928288195279676937966404069461, 3.07916926871191843506904711816, 4.86622130777510368577436397152, 6.39714317555827353831646835322, 7.49354775321479957545988309214, 8.839837742895997453616951799801, 9.640763673825080410958123124545, 10.21901100310839782682038709612, 11.60092514191548874754994905469, 12.94078456086323210845634924299
