| L(s) = 1 | − 13.6·3-s + 625·5-s − 6.44e3·7-s − 1.94e4·9-s + 1.08e4·11-s + 1.09e5·13-s − 8.55e3·15-s + 1.45e5·17-s + 1.88e5·19-s + 8.81e4·21-s − 2.57e6·23-s + 3.90e5·25-s + 5.36e5·27-s + 5.07e6·29-s − 2.12e6·31-s − 1.48e5·33-s − 4.02e6·35-s + 1.96e7·37-s − 1.49e6·39-s + 1.38e7·41-s + 1.50e7·43-s − 1.21e7·45-s + 2.89e7·47-s + 1.14e6·49-s − 1.98e6·51-s + 3.08e7·53-s + 6.78e6·55-s + ⋯ |
| L(s) = 1 | − 0.0975·3-s + 0.447·5-s − 1.01·7-s − 0.990·9-s + 0.223·11-s + 1.05·13-s − 0.0436·15-s + 0.421·17-s + 0.332·19-s + 0.0989·21-s − 1.91·23-s + 0.200·25-s + 0.194·27-s + 1.33·29-s − 0.412·31-s − 0.0218·33-s − 0.453·35-s + 1.72·37-s − 0.103·39-s + 0.764·41-s + 0.672·43-s − 0.442·45-s + 0.865·47-s + 0.0283·49-s − 0.0411·51-s + 0.536·53-s + 0.0999·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.718762288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.718762288\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 13.6T + 1.96e4T^{2} \) |
| 7 | \( 1 + 6.44e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.08e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.09e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.45e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.88e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.57e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.07e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.12e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.96e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.38e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.50e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.89e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.08e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.41e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.19e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.43e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.73e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.00e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.08e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.13e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.57e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.23e8T + 7.60e17T^{2} \) |
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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
−12.53847733474035010779660251867, −11.46667114965433620795217609749, −10.23917861728535073896032419912, −9.245326409519962045678519061592, −8.081637652255168044024745732030, −6.39135352349608120122047453483, −5.73498748580020117360238736623, −3.84155883227607337060474503051, −2.56289362614638317153958130361, −0.76689926686129540612453069092,
0.76689926686129540612453069092, 2.56289362614638317153958130361, 3.84155883227607337060474503051, 5.73498748580020117360238736623, 6.39135352349608120122047453483, 8.081637652255168044024745732030, 9.245326409519962045678519061592, 10.23917861728535073896032419912, 11.46667114965433620795217609749, 12.53847733474035010779660251867
