| L(s) = 1 | − 2.61·2-s − 0.381·3-s + 4.85·4-s − 3·5-s + 6-s − 3.85·7-s − 7.47·8-s − 2.85·9-s + 7.85·10-s + 2.23·11-s − 1.85·12-s − 0.145·13-s + 10.0·14-s + 1.14·15-s + 9.85·16-s − 0.763·17-s + 7.47·18-s − 2.85·19-s − 14.5·20-s + 1.47·21-s − 5.85·22-s + 7.47·23-s + 2.85·24-s + 4·25-s + 0.381·26-s + 2.23·27-s − 18.7·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 0.220·3-s + 2.42·4-s − 1.34·5-s + 0.408·6-s − 1.45·7-s − 2.64·8-s − 0.951·9-s + 2.48·10-s + 0.674·11-s − 0.535·12-s − 0.0404·13-s + 2.69·14-s + 0.295·15-s + 2.46·16-s − 0.185·17-s + 1.76·18-s − 0.654·19-s − 3.25·20-s + 0.321·21-s − 1.24·22-s + 1.55·23-s + 0.582·24-s + 0.800·25-s + 0.0749·26-s + 0.430·27-s − 3.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 67 | \( 1 + T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + 0.381T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 0.145T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 + 2.85T + 19T^{2} \) |
| 23 | \( 1 - 7.47T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 3.85T + 37T^{2} \) |
| 41 | \( 1 + 0.381T + 41T^{2} \) |
| 43 | \( 1 - 4.85T + 43T^{2} \) |
| 47 | \( 1 + 8.61T + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 71 | \( 1 - 3.76T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 6.56T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 2.23T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−14.91523776608311472957591545000, −12.67161903629462888514149979353, −11.56001067041141741538524132919, −10.85651290875254975429531714035, −9.419204322289150007108477222310, −8.640881423482157646777216209235, −7.35273556158538514937136442147, −6.36470697660250855000839936046, −3.25253030362544283348529857708, 0,
3.25253030362544283348529857708, 6.36470697660250855000839936046, 7.35273556158538514937136442147, 8.640881423482157646777216209235, 9.419204322289150007108477222310, 10.85651290875254975429531714035, 11.56001067041141741538524132919, 12.67161903629462888514149979353, 14.91523776608311472957591545000
