| L(s) = 1 | + 201.·3-s + 1.70e3·5-s + 2.40e3·7-s + 2.09e4·9-s + 5.46e4·11-s − 2.83e4·13-s + 3.43e5·15-s + 2.32e5·17-s − 9.40e5·19-s + 4.83e5·21-s + 1.11e6·23-s + 9.45e5·25-s + 2.47e5·27-s − 6.85e5·29-s − 8.63e6·31-s + 1.10e7·33-s + 4.08e6·35-s + 8.24e6·37-s − 5.71e6·39-s + 9.92e6·41-s + 3.26e7·43-s + 3.56e7·45-s + 5.89e7·47-s + 5.76e6·49-s + 4.68e7·51-s − 1.78e7·53-s + 9.31e7·55-s + ⋯ |
| L(s) = 1 | + 1.43·3-s + 1.21·5-s + 0.377·7-s + 1.06·9-s + 1.12·11-s − 0.275·13-s + 1.74·15-s + 0.675·17-s − 1.65·19-s + 0.542·21-s + 0.831·23-s + 0.483·25-s + 0.0896·27-s − 0.179·29-s − 1.67·31-s + 1.61·33-s + 0.460·35-s + 0.723·37-s − 0.395·39-s + 0.548·41-s + 1.45·43-s + 1.29·45-s + 1.76·47-s + 0.142·49-s + 0.970·51-s − 0.311·53-s + 1.37·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.324610106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.324610106\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 3 | \( 1 - 201.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.70e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 5.46e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.83e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.32e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.40e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.11e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.85e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.63e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.24e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 9.92e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.26e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.89e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.78e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 5.88e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.57e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.98e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.80e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.57e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.43e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.71e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.90e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.14e9T + 7.60e17T^{2} \) |
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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
−13.58317385085801402476329860576, −12.54989806531601116662802813011, −10.76411009453038077894338250494, −9.403613159711551110145161074814, −8.870861653097992729055624387131, −7.42423089851295923849544395524, −5.91589514745588508270350897637, −4.08156061254797022888569173525, −2.53294778499738433501055480949, −1.52239516414818728958455560929,
1.52239516414818728958455560929, 2.53294778499738433501055480949, 4.08156061254797022888569173525, 5.91589514745588508270350897637, 7.42423089851295923849544395524, 8.870861653097992729055624387131, 9.403613159711551110145161074814, 10.76411009453038077894338250494, 12.54989806531601116662802813011, 13.58317385085801402476329860576
