| L(s) = 1 | − 41.3·2-s + 1.19e3·4-s + 5.31e3·7-s − 2.82e4·8-s − 1.04e4·11-s + 7.96e4·13-s − 2.19e5·14-s + 5.54e5·16-s + 3.13e5·17-s − 2.46e5·19-s + 4.30e5·22-s + 7.21e5·23-s − 3.29e6·26-s + 6.35e6·28-s − 2.56e6·29-s − 3.29e6·31-s − 8.45e6·32-s − 1.29e7·34-s − 1.40e7·37-s + 1.02e7·38-s − 1.70e7·41-s − 2.92e7·43-s − 1.24e7·44-s − 2.98e7·46-s + 4.10e7·47-s − 1.21e7·49-s + 9.52e7·52-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.33·4-s + 0.836·7-s − 2.43·8-s − 0.214·11-s + 0.773·13-s − 1.52·14-s + 2.11·16-s + 0.911·17-s − 0.434·19-s + 0.392·22-s + 0.537·23-s − 1.41·26-s + 1.95·28-s − 0.674·29-s − 0.640·31-s − 1.42·32-s − 1.66·34-s − 1.23·37-s + 0.793·38-s − 0.941·41-s − 1.30·43-s − 0.501·44-s − 0.982·46-s + 1.22·47-s − 0.299·49-s + 1.80·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 41.3T + 512T^{2} \) |
| 7 | \( 1 - 5.31e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.04e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 7.96e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.13e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.46e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 7.21e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.56e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.29e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.40e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.70e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.92e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.10e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.67e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.60e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.33e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.80e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 8.97e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 7.60e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.10e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.21e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.37e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.03e8T + 7.60e17T^{2} \) |
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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
−10.07233648604065477063451206086, −9.026132108535508420138980357319, −8.289827428573339907474999239281, −7.54383171606175747810759929710, −6.51334377817221882245833951966, −5.24252358963507398813168833454, −3.40473647467303607998386882996, −1.96195354948829027850541244021, −1.20865845156661094730208039646, 0,
1.20865845156661094730208039646, 1.96195354948829027850541244021, 3.40473647467303607998386882996, 5.24252358963507398813168833454, 6.51334377817221882245833951966, 7.54383171606175747810759929710, 8.289827428573339907474999239281, 9.026132108535508420138980357319, 10.07233648604065477063451206086
