| L(s) = 1 | − 0.0532·2-s + 3-s − 1.99·4-s − 3.85·5-s − 0.0532·6-s + 4.59·7-s + 0.212·8-s + 9-s + 0.205·10-s + 2.84·11-s − 1.99·12-s + 5.18·13-s − 0.244·14-s − 3.85·15-s + 3.98·16-s − 3.00·17-s − 0.0532·18-s − 1.00·19-s + 7.69·20-s + 4.59·21-s − 0.151·22-s + 6.96·23-s + 0.212·24-s + 9.86·25-s − 0.276·26-s + 27-s − 9.16·28-s + ⋯ |
| L(s) = 1 | − 0.0376·2-s + 0.577·3-s − 0.998·4-s − 1.72·5-s − 0.0217·6-s + 1.73·7-s + 0.0753·8-s + 0.333·9-s + 0.0649·10-s + 0.857·11-s − 0.576·12-s + 1.43·13-s − 0.0653·14-s − 0.995·15-s + 0.995·16-s − 0.729·17-s − 0.0125·18-s − 0.231·19-s + 1.72·20-s + 1.00·21-s − 0.0323·22-s + 1.45·23-s + 0.0434·24-s + 1.97·25-s − 0.0542·26-s + 0.192·27-s − 1.73·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.263657067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.263657067\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0532T + 2T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 - 4.59T + 7T^{2} \) |
| 11 | \( 1 - 2.84T + 11T^{2} \) |
| 13 | \( 1 - 5.18T + 13T^{2} \) |
| 17 | \( 1 + 3.00T + 17T^{2} \) |
| 19 | \( 1 + 1.00T + 19T^{2} \) |
| 23 | \( 1 - 6.96T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 + 8.71T + 31T^{2} \) |
| 37 | \( 1 + 0.247T + 37T^{2} \) |
| 41 | \( 1 + 9.02T + 41T^{2} \) |
| 43 | \( 1 - 2.73T + 43T^{2} \) |
| 47 | \( 1 + 6.45T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 0.912T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 - 0.554T + 67T^{2} \) |
| 71 | \( 1 + 4.61T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 4.03T + 79T^{2} \) |
| 83 | \( 1 - 4.63T + 83T^{2} \) |
| 89 | \( 1 + 4.31T + 89T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
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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
−11.28077946690711357136009222119, −10.75231253696416012507490884172, −8.939859643084744026016181677193, −8.605799534098969891307087955041, −7.968388493951098915900269405606, −6.95088285854969722662398419613, −5.05917824729528854788224209293, −4.23050702813923280241919522805, −3.57381013585231205127714191766, −1.23071526684347095666156959402,
1.23071526684347095666156959402, 3.57381013585231205127714191766, 4.23050702813923280241919522805, 5.05917824729528854788224209293, 6.95088285854969722662398419613, 7.968388493951098915900269405606, 8.605799534098969891307087955041, 8.939859643084744026016181677193, 10.75231253696416012507490884172, 11.28077946690711357136009222119
