| L(s) = 1 | − 5.27·2-s − 4.20·4-s + 17.5·5-s + 172.·7-s + 190.·8-s − 92.4·10-s − 249.·11-s + 282.·13-s − 910.·14-s − 871.·16-s + 1.34e3·17-s − 2.88e3·19-s − 73.7·20-s + 1.31e3·22-s − 692.·23-s − 2.81e3·25-s − 1.49e3·26-s − 726.·28-s + 2.77e3·29-s − 4.10e3·31-s − 1.51e3·32-s − 7.10e3·34-s + 3.02e3·35-s − 1.06e4·37-s + 1.52e4·38-s + 3.34e3·40-s + 828.·41-s + ⋯ |
| L(s) = 1 | − 0.931·2-s − 0.131·4-s + 0.313·5-s + 1.33·7-s + 1.05·8-s − 0.292·10-s − 0.621·11-s + 0.464·13-s − 1.24·14-s − 0.851·16-s + 1.13·17-s − 1.83·19-s − 0.0412·20-s + 0.579·22-s − 0.273·23-s − 0.901·25-s − 0.432·26-s − 0.175·28-s + 0.613·29-s − 0.767·31-s − 0.261·32-s − 1.05·34-s + 0.417·35-s − 1.27·37-s + 1.70·38-s + 0.330·40-s + 0.0769·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 + 5.27T + 32T^{2} \) |
| 5 | \( 1 - 17.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 172.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 249.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 282.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.34e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.88e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 692.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.10e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.06e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 828.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.99e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.34e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.79e3T + 4.18e8T^{2} \) |
| 61 | \( 1 + 2.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.94e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.77e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.75e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.15e5T + 8.58e9T^{2} \) |
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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
−9.582905841235381341672278606332, −8.517911176685381825186492876622, −8.150730643205486648729280310907, −7.26415146173668338115190969538, −5.85443206166009173858608770642, −4.91876512179724410716204020893, −3.94877816257789115017265338294, −2.15028238152495995567381811869, −1.31767614426222118052860088612, 0,
1.31767614426222118052860088612, 2.15028238152495995567381811869, 3.94877816257789115017265338294, 4.91876512179724410716204020893, 5.85443206166009173858608770642, 7.26415146173668338115190969538, 8.150730643205486648729280310907, 8.517911176685381825186492876622, 9.582905841235381341672278606332
