| L(s) = 1 | − 3-s + 2.93·5-s + 2.71·7-s + 9-s − 3.58·11-s − 3.17·13-s − 2.93·15-s + 4.47·17-s − 2.55·19-s − 2.71·21-s − 23-s + 3.61·25-s − 27-s − 29-s − 5.38·31-s + 3.58·33-s + 7.96·35-s + 5.07·37-s + 3.17·39-s + 0.235·41-s + 8.18·43-s + 2.93·45-s − 6.76·47-s + 0.370·49-s − 4.47·51-s − 2.07·53-s − 10.5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.31·5-s + 1.02·7-s + 0.333·9-s − 1.08·11-s − 0.879·13-s − 0.757·15-s + 1.08·17-s − 0.586·19-s − 0.592·21-s − 0.208·23-s + 0.723·25-s − 0.192·27-s − 0.185·29-s − 0.966·31-s + 0.624·33-s + 1.34·35-s + 0.834·37-s + 0.507·39-s + 0.0368·41-s + 1.24·43-s + 0.437·45-s − 0.987·47-s + 0.0529·49-s − 0.627·51-s − 0.284·53-s − 1.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.212560972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.212560972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 2.93T + 5T^{2} \) |
| 7 | \( 1 - 2.71T + 7T^{2} \) |
| 11 | \( 1 + 3.58T + 11T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 2.55T + 19T^{2} \) |
| 31 | \( 1 + 5.38T + 31T^{2} \) |
| 37 | \( 1 - 5.07T + 37T^{2} \) |
| 41 | \( 1 - 0.235T + 41T^{2} \) |
| 43 | \( 1 - 8.18T + 43T^{2} \) |
| 47 | \( 1 + 6.76T + 47T^{2} \) |
| 53 | \( 1 + 2.07T + 53T^{2} \) |
| 59 | \( 1 - 5.67T + 59T^{2} \) |
| 61 | \( 1 - 3.60T + 61T^{2} \) |
| 67 | \( 1 - 2.30T + 67T^{2} \) |
| 71 | \( 1 - 5.54T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 4.16T + 79T^{2} \) |
| 83 | \( 1 - 6.64T + 83T^{2} \) |
| 89 | \( 1 - 8.84T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−7.75583949294436440268673023196, −7.23276850115234079714662484633, −6.20639103119684099444558175544, −5.71435084313554376099946119397, −5.07536503756758602561480441150, −4.69977358485372387118030944438, −3.48374686520687359804512856681, −2.29413696318705513300600210742, −1.94998879186240629938894508721, −0.75527539914700222677692907171,
0.75527539914700222677692907171, 1.94998879186240629938894508721, 2.29413696318705513300600210742, 3.48374686520687359804512856681, 4.69977358485372387118030944438, 5.07536503756758602561480441150, 5.71435084313554376099946119397, 6.20639103119684099444558175544, 7.23276850115234079714662484633, 7.75583949294436440268673023196
