| L(s) = 1 | + 272.·3-s + 625·5-s − 1.00e4·7-s + 5.44e4·9-s − 4.70e4·11-s − 9.36e3·13-s + 1.70e5·15-s + 1.08e5·17-s + 6.65e5·19-s − 2.72e6·21-s + 5.76e5·23-s + 3.90e5·25-s + 9.45e6·27-s + 2.61e6·29-s + 3.87e6·31-s − 1.28e7·33-s − 6.25e6·35-s − 1.41e7·37-s − 2.54e6·39-s + 4.62e6·41-s − 8.31e6·43-s + 3.40e7·45-s + 2.51e7·47-s + 5.96e7·49-s + 2.95e7·51-s − 3.49e7·53-s − 2.94e7·55-s + ⋯ |
| L(s) = 1 | + 1.94·3-s + 0.447·5-s − 1.57·7-s + 2.76·9-s − 0.969·11-s − 0.0909·13-s + 0.867·15-s + 0.315·17-s + 1.17·19-s − 3.05·21-s + 0.429·23-s + 0.200·25-s + 3.42·27-s + 0.685·29-s + 0.754·31-s − 1.88·33-s − 0.704·35-s − 1.24·37-s − 0.176·39-s + 0.255·41-s − 0.370·43-s + 1.23·45-s + 0.753·47-s + 1.47·49-s + 0.611·51-s − 0.608·53-s − 0.433·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.811367440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.811367440\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 625T \) |
| good | 3 | \( 1 - 272.T + 1.96e4T^{2} \) |
| 7 | \( 1 + 1.00e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.70e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.36e3T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.08e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.65e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 5.76e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.61e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.87e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.41e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.62e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 8.31e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.51e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.49e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.71e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 4.75e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.38e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.54e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.41e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.61e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.55e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.00e9T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.24e9T + 7.60e17T^{2} \) |
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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
−9.806111092093206698628367134793, −9.258830798952659589816318983697, −8.290640060241813325531201932617, −7.38513041624100549502056956684, −6.54091233503484766916107835264, −5.05510438497961833948528272625, −3.58490534685648751845402052716, −3.01125816817505698124878392717, −2.25114601932112852843857574340, −0.866211536703653117624254502041,
0.866211536703653117624254502041, 2.25114601932112852843857574340, 3.01125816817505698124878392717, 3.58490534685648751845402052716, 5.05510438497961833948528272625, 6.54091233503484766916107835264, 7.38513041624100549502056956684, 8.290640060241813325531201932617, 9.258830798952659589816318983697, 9.806111092093206698628367134793
