| L(s) = 1 | − 18.1·2-s + 81·3-s − 181.·4-s + 1.09e3·5-s − 1.47e3·6-s − 2.88e3·7-s + 1.26e4·8-s + 6.56e3·9-s − 1.99e4·10-s + 1.67e3·11-s − 1.47e4·12-s − 1.28e4·13-s + 5.24e4·14-s + 8.88e4·15-s − 1.35e5·16-s − 1.00e5·17-s − 1.19e5·18-s − 5.09e5·19-s − 1.99e5·20-s − 2.33e5·21-s − 3.04e4·22-s + 2.00e6·23-s + 1.02e6·24-s − 7.50e5·25-s + 2.33e5·26-s + 5.31e5·27-s + 5.24e5·28-s + ⋯ |
| L(s) = 1 | − 0.803·2-s + 0.577·3-s − 0.355·4-s + 0.784·5-s − 0.463·6-s − 0.454·7-s + 1.08·8-s + 0.333·9-s − 0.630·10-s + 0.0345·11-s − 0.205·12-s − 0.124·13-s + 0.364·14-s + 0.452·15-s − 0.518·16-s − 0.292·17-s − 0.267·18-s − 0.896·19-s − 0.278·20-s − 0.262·21-s − 0.0277·22-s + 1.49·23-s + 0.628·24-s − 0.384·25-s + 0.100·26-s + 0.192·27-s + 0.161·28-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
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 + 18.1T + 512T^{2} \) |
| 5 | \( 1 - 1.09e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 2.88e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.67e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.28e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.00e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.09e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.00e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.00e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.60e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.57e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.77e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.57e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.42e5T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.97e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 7.35e6T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.33e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.69e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.14e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.11e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.77e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.71e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.38e9T + 7.60e17T^{2} \) |
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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
−10.16567359769265209029679001725, −9.369664084830197404481589073395, −8.791968337547547623506009443640, −7.66172301312610357510563186988, −6.56606740318991232504455517282, −5.16973462088667256984890951653, −3.90481829752719876303416520200, −2.44828828667770646848868843969, −1.33288465611583035828799815055, 0,
1.33288465611583035828799815055, 2.44828828667770646848868843969, 3.90481829752719876303416520200, 5.16973462088667256984890951653, 6.56606740318991232504455517282, 7.66172301312610357510563186988, 8.791968337547547623506009443640, 9.369664084830197404481589073395, 10.16567359769265209029679001725
