| L(s) = 1 | + 1.33·2-s − 81·3-s − 510.·4-s + 2.09e3·5-s − 108.·6-s + 107.·7-s − 1.36e3·8-s + 6.56e3·9-s + 2.79e3·10-s − 839.·11-s + 4.13e4·12-s + 3.40e4·13-s + 143.·14-s − 1.69e5·15-s + 2.59e5·16-s − 8.47e4·17-s + 8.76e3·18-s + 9.28e5·19-s − 1.06e6·20-s − 8.68e3·21-s − 1.12e3·22-s − 9.71e5·23-s + 1.10e5·24-s + 2.41e6·25-s + 4.54e4·26-s − 5.31e5·27-s − 5.46e4·28-s + ⋯ |
| L(s) = 1 | + 0.0590·2-s − 0.577·3-s − 0.996·4-s + 1.49·5-s − 0.0340·6-s + 0.0168·7-s − 0.117·8-s + 0.333·9-s + 0.0883·10-s − 0.0172·11-s + 0.575·12-s + 0.330·13-s + 0.000996·14-s − 0.863·15-s + 0.989·16-s − 0.246·17-s + 0.0196·18-s + 1.63·19-s − 1.49·20-s − 0.00973·21-s − 0.00102·22-s − 0.724·23-s + 0.0680·24-s + 1.23·25-s + 0.0195·26-s − 0.192·27-s − 0.0168·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.922173662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.922173662\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 - 1.33T + 512T^{2} \) |
| 5 | \( 1 - 2.09e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 107.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 839.T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.40e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 8.47e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.28e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 9.71e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.34e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.99e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.60e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.18e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.85e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.01e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.57e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 6.03e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.83e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.57e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.74e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 6.00e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.16e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.58e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.10e9T + 7.60e17T^{2} \) |
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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
−10.85075274353101980426840779772, −9.778600509661959303507539642536, −9.392625635194499930440013086177, −8.075742564046768143576225432036, −6.57080046881243057040588183662, −5.58258930168073430595451918747, −4.95024859552373340287748391903, −3.46650583855486516774469102380, −1.83207216206507510058882247409, −0.72863430862231156100194204877,
0.72863430862231156100194204877, 1.83207216206507510058882247409, 3.46650583855486516774469102380, 4.95024859552373340287748391903, 5.58258930168073430595451918747, 6.57080046881243057040588183662, 8.075742564046768143576225432036, 9.392625635194499930440013086177, 9.778600509661959303507539642536, 10.85075274353101980426840779772
