| L(s) = 1 | + 18.4·3-s + 25·5-s + 121.·7-s + 98.1·9-s + 438.·11-s + 758.·13-s + 461.·15-s − 1.53e3·17-s − 75.8·19-s + 2.25e3·21-s + 3.69e3·23-s + 625·25-s − 2.67e3·27-s − 6.32e3·29-s + 2.69e3·31-s + 8.09e3·33-s + 3.04e3·35-s + 7.25e3·37-s + 1.40e4·39-s + 4.91e3·41-s − 2.53e3·43-s + 2.45e3·45-s + 1.13e4·47-s − 1.94e3·49-s − 2.83e4·51-s − 2.94e4·53-s + 1.09e4·55-s + ⋯ |
| L(s) = 1 | + 1.18·3-s + 0.447·5-s + 0.940·7-s + 0.404·9-s + 1.09·11-s + 1.24·13-s + 0.529·15-s − 1.28·17-s − 0.0482·19-s + 1.11·21-s + 1.45·23-s + 0.200·25-s − 0.706·27-s − 1.39·29-s + 0.502·31-s + 1.29·33-s + 0.420·35-s + 0.870·37-s + 1.47·39-s + 0.456·41-s − 0.208·43-s + 0.180·45-s + 0.751·47-s − 0.115·49-s − 1.52·51-s − 1.43·53-s + 0.488·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.319842565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.319842565\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 3 | \( 1 - 18.4T + 243T^{2} \) |
| 7 | \( 1 - 121.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 438.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 758.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.53e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 75.8T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.25e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.91e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.53e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.13e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.94e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.68e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.95e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.79e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.12e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.69e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.31e5T + 8.58e9T^{2} \) |
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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
−11.03067228187167203717958131934, −9.469964174139763375517953156896, −8.890103223737571750570377065123, −8.237197059493658494651205127485, −7.05268493215824338189375202288, −5.96069053823299166018170098387, −4.54108550092915366024449072680, −3.51017760547060839648702056210, −2.21950048554709221273862364784, −1.23730372781136489716802478269,
1.23730372781136489716802478269, 2.21950048554709221273862364784, 3.51017760547060839648702056210, 4.54108550092915366024449072680, 5.96069053823299166018170098387, 7.05268493215824338189375202288, 8.237197059493658494651205127485, 8.890103223737571750570377065123, 9.469964174139763375517953156896, 11.03067228187167203717958131934
