| L(s) = 1 | + 24.6·2-s − 81·3-s + 96.9·4-s + 1.35e3·5-s − 1.99e3·6-s − 4.83e3·7-s − 1.02e4·8-s + 6.56e3·9-s + 3.33e4·10-s + 1.73e4·11-s − 7.85e3·12-s + 1.56e5·13-s − 1.19e5·14-s − 1.09e5·15-s − 3.02e5·16-s − 3.58e5·17-s + 1.61e5·18-s + 7.77e4·19-s + 1.31e5·20-s + 3.91e5·21-s + 4.29e5·22-s + 1.04e6·23-s + 8.29e5·24-s − 1.25e5·25-s + 3.85e6·26-s − 5.31e5·27-s − 4.68e5·28-s + ⋯ |
| L(s) = 1 | + 1.09·2-s − 0.577·3-s + 0.189·4-s + 0.967·5-s − 0.629·6-s − 0.760·7-s − 0.884·8-s + 0.333·9-s + 1.05·10-s + 0.358·11-s − 0.109·12-s + 1.51·13-s − 0.829·14-s − 0.558·15-s − 1.15·16-s − 1.04·17-s + 0.363·18-s + 0.136·19-s + 0.183·20-s + 0.439·21-s + 0.390·22-s + 0.775·23-s + 0.510·24-s − 0.0644·25-s + 1.65·26-s − 0.192·27-s − 0.143·28-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 | \( 1 - 24.6T + 512T^{2} \) |
| 5 | \( 1 - 1.35e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.83e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.73e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.56e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.58e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.77e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.04e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.51e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.81e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 5.36e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.59e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.04e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.40e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.96e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.50e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.05e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 4.08e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.83e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.66e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.10e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.56e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.46e8T + 7.60e17T^{2} \) |
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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
−10.72390985534786195440482374018, −9.514157089072407277559305759201, −8.756989721892459505815895470990, −6.69561359933512112306909021490, −6.16645587966291002421469605125, −5.28549596830170248841850180847, −4.09824536444775189461111915728, −3.04352096199006403054110174327, −1.52103987204814498548091550181, 0,
1.52103987204814498548091550181, 3.04352096199006403054110174327, 4.09824536444775189461111915728, 5.28549596830170248841850180847, 6.16645587966291002421469605125, 6.69561359933512112306909021490, 8.756989721892459505815895470990, 9.514157089072407277559305759201, 10.72390985534786195440482374018
