| L(s) = 1 | − 2.24e3·2-s + 5.90e4·3-s + 2.92e6·4-s − 1.32e8·6-s − 1.14e8·7-s − 1.85e9·8-s + 3.48e9·9-s − 8.03e10·11-s + 1.72e11·12-s + 2.07e11·13-s + 2.56e11·14-s − 1.97e12·16-s + 2.89e12·17-s − 7.81e12·18-s − 1.26e13·19-s − 6.74e12·21-s + 1.80e14·22-s − 1.21e14·23-s − 1.09e14·24-s − 4.63e14·26-s + 2.05e14·27-s − 3.34e14·28-s + 2.61e15·29-s + 6.16e15·31-s + 8.31e15·32-s − 4.74e15·33-s − 6.48e15·34-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 0.577·3-s + 1.39·4-s − 0.893·6-s − 0.152·7-s − 0.611·8-s + 0.333·9-s − 0.934·11-s + 0.805·12-s + 0.416·13-s + 0.236·14-s − 0.448·16-s + 0.347·17-s − 0.515·18-s − 0.473·19-s − 0.0882·21-s + 1.44·22-s − 0.612·23-s − 0.353·24-s − 0.644·26-s + 0.192·27-s − 0.213·28-s + 1.15·29-s + 1.35·31-s + 1.30·32-s − 0.539·33-s − 0.538·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.24e3T + 2.09e6T^{2} \) |
| 7 | \( 1 + 1.14e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 8.03e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 2.07e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 2.89e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.26e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 1.21e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.61e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 6.16e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 4.84e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.97e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 2.05e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 3.51e16T + 1.30e35T^{2} \) |
| 53 | \( 1 + 5.84e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 8.39e17T + 1.54e37T^{2} \) |
| 61 | \( 1 - 3.71e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.38e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 4.03e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 2.77e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 2.28e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.43e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 3.55e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.26e21T + 5.27e41T^{2} \) |
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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
−10.08025981982625862428976134976, −8.900358745772192496797114723141, −8.209976919774533191597486183684, −7.37516843543398011501884849049, −6.19442045429740231411469409587, −4.56897677887037087755012082105, −3.05728523500351391366237151942, −2.08894718138870455243546349762, −1.03103409658108906459693530120, 0,
1.03103409658108906459693530120, 2.08894718138870455243546349762, 3.05728523500351391366237151942, 4.56897677887037087755012082105, 6.19442045429740231411469409587, 7.37516843543398011501884849049, 8.209976919774533191597486183684, 8.900358745772192496797114723141, 10.08025981982625862428976134976
