| L(s) = 1 | + 3-s + 3.43·5-s + 2.70·7-s + 9-s − 11-s − 13-s + 3.43·15-s − 0.375·17-s + 2.70·19-s + 2.70·21-s + 0.938·23-s + 6.81·25-s + 27-s + 0.728·29-s − 6.85·31-s − 33-s + 9.30·35-s + 9.89·37-s − 39-s − 2.83·41-s − 1.62·43-s + 3.43·45-s − 1.10·47-s + 0.334·49-s − 0.375·51-s + 12.1·53-s − 3.43·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.53·5-s + 1.02·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s + 0.887·15-s − 0.0911·17-s + 0.621·19-s + 0.591·21-s + 0.195·23-s + 1.36·25-s + 0.192·27-s + 0.135·29-s − 1.23·31-s − 0.174·33-s + 1.57·35-s + 1.62·37-s − 0.160·39-s − 0.442·41-s − 0.247·43-s + 0.512·45-s − 0.161·47-s + 0.0478·49-s − 0.0526·51-s + 1.67·53-s − 0.463·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.146399659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.146399659\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.43T + 5T^{2} \) |
| 7 | \( 1 - 2.70T + 7T^{2} \) |
| 17 | \( 1 + 0.375T + 17T^{2} \) |
| 19 | \( 1 - 2.70T + 19T^{2} \) |
| 23 | \( 1 - 0.938T + 23T^{2} \) |
| 29 | \( 1 - 0.728T + 29T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 + 2.83T + 41T^{2} \) |
| 43 | \( 1 + 1.62T + 43T^{2} \) |
| 47 | \( 1 + 1.10T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 2.62T + 59T^{2} \) |
| 61 | \( 1 + 3.10T + 61T^{2} \) |
| 67 | \( 1 + 4.73T + 67T^{2} \) |
| 71 | \( 1 - 0.108T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 + 8.03T + 79T^{2} \) |
| 83 | \( 1 - 4.99T + 83T^{2} \) |
| 89 | \( 1 - 6.33T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−7.930361863872726977584544557320, −7.38974372988970453591924618057, −6.55303009132354066834507603108, −5.73668719900097654026298273973, −5.18378589719848449259635802950, −4.53411514219471754818187551590, −3.44804604564974339517074778603, −2.46983642489964295475084776557, −1.96559729497285989929488315774, −1.08740082906941530998716811507,
1.08740082906941530998716811507, 1.96559729497285989929488315774, 2.46983642489964295475084776557, 3.44804604564974339517074778603, 4.53411514219471754818187551590, 5.18378589719848449259635802950, 5.73668719900097654026298273973, 6.55303009132354066834507603108, 7.38974372988970453591924618057, 7.930361863872726977584544557320
