| L(s) = 1 | + 10.1·2-s − 16.4·3-s + 71.0·4-s − 99.7·5-s − 167.·6-s − 57.1·7-s + 396.·8-s + 27.8·9-s − 1.01e3·10-s − 438.·11-s − 1.16e3·12-s + 477.·13-s − 579.·14-s + 1.64e3·15-s + 1.75e3·16-s + 1.67e3·17-s + 282.·18-s − 7.08e3·20-s + 940.·21-s − 4.45e3·22-s + 459.·23-s − 6.52e3·24-s + 6.82e3·25-s + 4.84e3·26-s + 3.54e3·27-s − 4.05e3·28-s + 2.69e3·29-s + ⋯ |
| L(s) = 1 | + 1.79·2-s − 1.05·3-s + 2.22·4-s − 1.78·5-s − 1.89·6-s − 0.440·7-s + 2.19·8-s + 0.114·9-s − 3.20·10-s − 1.09·11-s − 2.34·12-s + 0.783·13-s − 0.790·14-s + 1.88·15-s + 1.71·16-s + 1.40·17-s + 0.205·18-s − 3.96·20-s + 0.465·21-s − 1.96·22-s + 0.181·23-s − 2.31·24-s + 2.18·25-s + 1.40·26-s + 0.934·27-s − 0.978·28-s + 0.595·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.529778311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.529778311\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 - 10.1T + 32T^{2} \) |
| 3 | \( 1 + 16.4T + 243T^{2} \) |
| 5 | \( 1 + 99.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 57.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 438.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 477.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.67e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 459.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.31e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.48e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.10e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.80e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.32e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.28e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.69e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.56e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.34e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.73e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.91e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.31e4T + 8.58e9T^{2} \) |
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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
−11.04555665283632947045640644488, −10.44068469983726278104731553043, −8.302795317079775991104484869985, −7.40227643657287077538578159210, −6.44904554838721600434452956992, −5.49346150516245078754529339785, −4.74955320962310450293106222654, −3.67510676812667779326799529235, −2.96474384039165576444206244868, −0.67684564049302868020312040009,
0.67684564049302868020312040009, 2.96474384039165576444206244868, 3.67510676812667779326799529235, 4.74955320962310450293106222654, 5.49346150516245078754529339785, 6.44904554838721600434452956992, 7.40227643657287077538578159210, 8.302795317079775991104484869985, 10.44068469983726278104731553043, 11.04555665283632947045640644488
