| L(s) = 1 | + 2.16e7·3-s + 6.04e11·5-s + 1.12e14·7-s − 5.08e15·9-s + 2.12e17·11-s − 2.20e18·13-s + 1.31e19·15-s − 9.09e19·17-s + 5.28e20·19-s + 2.43e21·21-s − 2.52e22·23-s + 2.49e23·25-s − 2.31e23·27-s + 2.22e24·29-s + 5.96e24·31-s + 4.59e24·33-s + 6.78e25·35-s − 1.07e25·37-s − 4.77e25·39-s + 5.45e25·41-s − 7.34e26·43-s − 3.07e27·45-s − 2.88e27·47-s + 4.86e27·49-s − 1.97e27·51-s − 6.58e27·53-s + 1.28e29·55-s + ⋯ |
| L(s) = 1 | + 0.290·3-s + 1.77·5-s + 1.27·7-s − 0.915·9-s + 1.39·11-s − 0.917·13-s + 0.515·15-s − 0.453·17-s + 0.420·19-s + 0.371·21-s − 0.858·23-s + 2.14·25-s − 0.557·27-s + 1.65·29-s + 1.47·31-s + 0.404·33-s + 2.26·35-s − 0.143·37-s − 0.266·39-s + 0.133·41-s − 0.820·43-s − 1.62·45-s − 0.741·47-s + 0.629·49-s − 0.131·51-s − 0.233·53-s + 2.46·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(\approx\) |
\(3.382012769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.382012769\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 2.16e7T + 5.55e15T^{2} \) |
| 5 | \( 1 - 6.04e11T + 1.16e23T^{2} \) |
| 7 | \( 1 - 1.12e14T + 7.73e27T^{2} \) |
| 11 | \( 1 - 2.12e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 2.20e18T + 5.75e36T^{2} \) |
| 17 | \( 1 + 9.09e19T + 4.02e40T^{2} \) |
| 19 | \( 1 - 5.28e20T + 1.58e42T^{2} \) |
| 23 | \( 1 + 2.52e22T + 8.65e44T^{2} \) |
| 29 | \( 1 - 2.22e24T + 1.81e48T^{2} \) |
| 31 | \( 1 - 5.96e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 1.07e25T + 5.63e51T^{2} \) |
| 41 | \( 1 - 5.45e25T + 1.66e53T^{2} \) |
| 43 | \( 1 + 7.34e26T + 8.02e53T^{2} \) |
| 47 | \( 1 + 2.88e27T + 1.51e55T^{2} \) |
| 53 | \( 1 + 6.58e27T + 7.96e56T^{2} \) |
| 59 | \( 1 + 1.55e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 5.35e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 5.20e29T + 1.82e60T^{2} \) |
| 71 | \( 1 - 2.97e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 5.09e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 1.81e30T + 4.18e62T^{2} \) |
| 83 | \( 1 + 3.91e31T + 2.13e63T^{2} \) |
| 89 | \( 1 + 1.76e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 8.13e32T + 3.65e65T^{2} \) |
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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
−17.19705859177165266343889893123, −14.46332418152513106338421879764, −13.93535191001986867513127557801, −11.73590723627168521649275674842, −9.890616235970197654756947968842, −8.549824988887918785174758742834, −6.34431902832542930260175433855, −4.90104549828883922952571876861, −2.48515890102930602818381678793, −1.36732755296961624989494891612,
1.36732755296961624989494891612, 2.48515890102930602818381678793, 4.90104549828883922952571876861, 6.34431902832542930260175433855, 8.549824988887918785174758742834, 9.890616235970197654756947968842, 11.73590723627168521649275674842, 13.93535191001986867513127557801, 14.46332418152513106338421879764, 17.19705859177165266343889893123
