| L(s) = 1 | − 0.414·2-s + 2.82·3-s − 1.82·4-s − 5-s − 1.17·6-s − 2·7-s + 1.58·8-s + 5.00·9-s + 0.414·10-s + 11-s − 5.17·12-s − 6.82·13-s + 0.828·14-s − 2.82·15-s + 3·16-s + 1.17·17-s − 2.07·18-s + 1.82·20-s − 5.65·21-s − 0.414·22-s + 2.82·23-s + 4.48·24-s + 25-s + 2.82·26-s + 5.65·27-s + 3.65·28-s + 7.65·29-s + ⋯ |
| L(s) = 1 | − 0.292·2-s + 1.63·3-s − 0.914·4-s − 0.447·5-s − 0.478·6-s − 0.755·7-s + 0.560·8-s + 1.66·9-s + 0.130·10-s + 0.301·11-s − 1.49·12-s − 1.89·13-s + 0.221·14-s − 0.730·15-s + 0.750·16-s + 0.284·17-s − 0.488·18-s + 0.408·20-s − 1.23·21-s − 0.0883·22-s + 0.589·23-s + 0.915·24-s + 0.200·25-s + 0.554·26-s + 1.08·27-s + 0.691·28-s + 1.42·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8899794316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8899794316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 3 | \( 1 - 2.82T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 + 6.82T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 0.343T + 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 4.48T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 9.31T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−14.94939385721676879893707388798, −14.35642538898121481045712364568, −13.25975961075160009500758146644, −12.32439160511051109383228422372, −10.01954456510364785732734865775, −9.373862471307683607328605854679, −8.288711442380031784730947830553, −7.24885125285007192371091012034, −4.53592171229370235630296301299, −3.03212513583617232663088011462,
3.03212513583617232663088011462, 4.53592171229370235630296301299, 7.24885125285007192371091012034, 8.288711442380031784730947830553, 9.373862471307683607328605854679, 10.01954456510364785732734865775, 12.32439160511051109383228422372, 13.25975961075160009500758146644, 14.35642538898121481045712364568, 14.94939385721676879893707388798
