| L(s) = 1 | + 3·3-s + 13.6·5-s + 15.6·7-s + 9·9-s + 50.5·11-s + 13·13-s + 40.8·15-s + 2·17-s − 18.1·19-s + 46.8·21-s + 64·23-s + 60.7·25-s + 27·27-s − 103.·29-s − 34.4·31-s + 151.·33-s + 213.·35-s − 267.·37-s + 39·39-s − 147.·41-s + 166.·43-s + 122.·45-s + 325.·47-s − 98.6·49-s + 6·51-s − 111.·53-s + 688.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.21·5-s + 0.843·7-s + 0.333·9-s + 1.38·11-s + 0.277·13-s + 0.703·15-s + 0.0285·17-s − 0.219·19-s + 0.487·21-s + 0.580·23-s + 0.486·25-s + 0.192·27-s − 0.664·29-s − 0.199·31-s + 0.799·33-s + 1.02·35-s − 1.18·37-s + 0.160·39-s − 0.561·41-s + 0.592·43-s + 0.406·45-s + 1.01·47-s − 0.287·49-s + 0.0164·51-s − 0.287·53-s + 1.68·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.847563012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.847563012\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 13.6T + 125T^{2} \) |
| 7 | \( 1 - 15.6T + 343T^{2} \) |
| 11 | \( 1 - 50.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 18.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 64T + 1.21e4T^{2} \) |
| 29 | \( 1 + 103.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 34.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 267.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 147.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 166.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 325.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 111.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 24.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 640.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 382.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 510.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 291.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 794.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 945.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 317.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.57e3T + 9.12e5T^{2} \) |
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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
−10.09761762605332434325181662931, −9.151614553780722590258903504894, −8.776302498750950502998872783074, −7.57218922446952078522760984649, −6.59817340332287382715892581792, −5.70370450698841604630706746641, −4.61936163324185044302251140777, −3.48484020754733292211297457744, −2.06223372633070112311336423289, −1.32977305060417499735762231754,
1.32977305060417499735762231754, 2.06223372633070112311336423289, 3.48484020754733292211297457744, 4.61936163324185044302251140777, 5.70370450698841604630706746641, 6.59817340332287382715892581792, 7.57218922446952078522760984649, 8.776302498750950502998872783074, 9.151614553780722590258903504894, 10.09761762605332434325181662931
