| L(s) = 1 | − 6.13·3-s + 2.36·5-s + 18.5·7-s + 10.6·9-s + 15.3·11-s + 27.8·13-s − 14.4·15-s − 62.4·17-s − 55.3·19-s − 113.·21-s − 44.3·23-s − 119.·25-s + 100.·27-s + 29·29-s + 207.·31-s − 94.3·33-s + 43.6·35-s + 303.·37-s − 170.·39-s + 125.·41-s − 101.·43-s + 25.0·45-s − 50.8·47-s − 0.583·49-s + 382.·51-s + 692.·53-s + 36.3·55-s + ⋯ |
| L(s) = 1 | − 1.18·3-s + 0.211·5-s + 0.999·7-s + 0.393·9-s + 0.421·11-s + 0.593·13-s − 0.249·15-s − 0.890·17-s − 0.668·19-s − 1.17·21-s − 0.401·23-s − 0.955·25-s + 0.716·27-s + 0.185·29-s + 1.19·31-s − 0.497·33-s + 0.210·35-s + 1.34·37-s − 0.700·39-s + 0.478·41-s − 0.359·43-s + 0.0829·45-s − 0.157·47-s − 0.00170·49-s + 1.05·51-s + 1.79·53-s + 0.0890·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.373565393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.373565393\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 6.13T + 27T^{2} \) |
| 5 | \( 1 - 2.36T + 125T^{2} \) |
| 7 | \( 1 - 18.5T + 343T^{2} \) |
| 11 | \( 1 - 15.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 62.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 55.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 44.3T + 1.21e4T^{2} \) |
| 31 | \( 1 - 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 303.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 125.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 101.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 50.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 692.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 557.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 809.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 749.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 54.7T + 3.57e5T^{2} \) |
| 73 | \( 1 + 184.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 752.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 902.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 953.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.11e3T + 9.12e5T^{2} \) |
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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
−10.89413929168215486096150024725, −9.954568727924472185916418233730, −8.753313449214129755316438224850, −7.946121515845367842464592195337, −6.59246006138717735587416733633, −5.99944551614285643960280578844, −4.94430085745551361012097073739, −4.08721955136850121331935798714, −2.17127880065591908178462290516, −0.799316347517982804200599575553,
0.799316347517982804200599575553, 2.17127880065591908178462290516, 4.08721955136850121331935798714, 4.94430085745551361012097073739, 5.99944551614285643960280578844, 6.59246006138717735587416733633, 7.946121515845367842464592195337, 8.753313449214129755316438224850, 9.954568727924472185916418233730, 10.89413929168215486096150024725
