| L(s) = 1 | − 0.618·2-s + 2.61·3-s − 1.61·4-s + 5-s − 1.61·6-s + 7-s + 2.23·8-s + 3.85·9-s − 0.618·10-s − 1.61·11-s − 4.23·12-s − 13-s − 0.618·14-s + 2.61·15-s + 1.85·16-s − 2.85·17-s − 2.38·18-s − 19-s − 1.61·20-s + 2.61·21-s + 1.00·22-s + 3.47·23-s + 5.85·24-s − 4·25-s + 0.618·26-s + 2.23·27-s − 1.61·28-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 1.51·3-s − 0.809·4-s + 0.447·5-s − 0.660·6-s + 0.377·7-s + 0.790·8-s + 1.28·9-s − 0.195·10-s − 0.487·11-s − 1.22·12-s − 0.277·13-s − 0.165·14-s + 0.675·15-s + 0.463·16-s − 0.692·17-s − 0.561·18-s − 0.229·19-s − 0.361·20-s + 0.571·21-s + 0.213·22-s + 0.723·23-s + 1.19·24-s − 0.800·25-s + 0.121·26-s + 0.430·27-s − 0.305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.224112906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.224112906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 2.85T + 17T^{2} \) |
| 23 | \( 1 - 3.47T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 0.472T + 43T^{2} \) |
| 47 | \( 1 + 1.47T + 47T^{2} \) |
| 53 | \( 1 + 1.85T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 5.94T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 4.70T + 71T^{2} \) |
| 73 | \( 1 - 4.32T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 9.85T + 83T^{2} \) |
| 89 | \( 1 - 7.23T + 89T^{2} \) |
| 97 | \( 1 + 1.47T + 97T^{2} \) |
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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
−13.45486519438234806445427367698, −12.71041944691866674160718347910, −10.85844929122760418066354542346, −9.766895861616560528521506473258, −8.994154382011203061738546809188, −8.262282164725696725643339386455, −7.25895705220832828808564338575, −5.19434758733118900016763246405, −3.79392358318710283826542288661, −2.12384877547305281217510389717,
2.12384877547305281217510389717, 3.79392358318710283826542288661, 5.19434758733118900016763246405, 7.25895705220832828808564338575, 8.262282164725696725643339386455, 8.994154382011203061738546809188, 9.766895861616560528521506473258, 10.85844929122760418066354542346, 12.71041944691866674160718347910, 13.45486519438234806445427367698
