| L(s) = 1 | + 32.7·2-s + 563.·4-s + 1.31e3·5-s + 5.15e3·7-s + 1.69e3·8-s + 4.31e4·10-s − 1.51e4·11-s + 1.80e5·13-s + 1.69e5·14-s − 2.33e5·16-s + 5.95e5·17-s − 7.85e5·19-s + 7.41e5·20-s − 4.96e5·22-s − 1.17e6·23-s − 2.22e5·25-s + 5.91e6·26-s + 2.90e6·28-s + 1.54e6·29-s − 2.27e6·31-s − 8.51e6·32-s + 1.95e7·34-s + 6.78e6·35-s − 1.12e7·37-s − 2.57e7·38-s + 2.22e6·40-s − 1.67e7·41-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + 1.10·4-s + 0.941·5-s + 0.812·7-s + 0.146·8-s + 1.36·10-s − 0.311·11-s + 1.75·13-s + 1.17·14-s − 0.889·16-s + 1.73·17-s − 1.38·19-s + 1.03·20-s − 0.451·22-s − 0.875·23-s − 0.114·25-s + 2.54·26-s + 0.893·28-s + 0.405·29-s − 0.441·31-s − 1.43·32-s + 2.50·34-s + 0.764·35-s − 0.990·37-s − 2.00·38-s + 0.137·40-s − 0.924·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.640756741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.640756741\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 32.7T + 512T^{2} \) |
| 5 | \( 1 - 1.31e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 5.15e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.51e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.80e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.95e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.85e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.17e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.54e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.27e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.12e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.67e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.11e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.93e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.77e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.37e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.84e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.43e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.34e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.57e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.69e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.45e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.10e9T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.96e8T + 7.60e17T^{2} \) |
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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
−14.79568822456169096969465434855, −13.92666281933895964797914930467, −13.07486950324238025596870580072, −11.75237727851274496538554588559, −10.36306072555061274921676127162, −8.425473719299466482171402438826, −6.25879986545126302621323228229, −5.28900321543201426210448343409, −3.66208161745196348406967059103, −1.79777596707653475781068263878,
1.79777596707653475781068263878, 3.66208161745196348406967059103, 5.28900321543201426210448343409, 6.25879986545126302621323228229, 8.425473719299466482171402438826, 10.36306072555061274921676127162, 11.75237727851274496538554588559, 13.07486950324238025596870580072, 13.92666281933895964797914930467, 14.79568822456169096969465434855
