| L(s) = 1 | − 19.1·3-s + (−55.2 − 8.66i)5-s − 121. i·7-s + 123.·9-s + 298i·11-s + 485.·13-s + (1.05e3 + 165. i)15-s − 420. i·17-s + 2.57e3i·19-s + 2.32e3i·21-s − 717. i·23-s + (2.97e3 + 956. i)25-s + 2.29e3·27-s + 3.73e3i·29-s − 6.22e3·31-s + ⋯ |
| L(s) = 1 | − 1.22·3-s + (−0.987 − 0.154i)5-s − 0.937i·7-s + 0.506·9-s + 0.742i·11-s + 0.797·13-s + (1.21 + 0.190i)15-s − 0.353i·17-s + 1.63i·19-s + 1.15i·21-s − 0.282i·23-s + (0.952 + 0.306i)25-s + 0.606·27-s + 0.824i·29-s − 1.16·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4584274146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4584274146\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (55.2 + 8.66i)T \) |
| good | 3 | \( 1 + 19.1T + 243T^{2} \) |
| 7 | \( 1 + 121. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 298iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 485.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 420. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.57e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 717. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.73e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 6.22e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.13e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.91e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.28e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.63e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 4.87e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 8.39e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.90e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 420. iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.11e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.06e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.80e5iT - 8.58e9T^{2} \) |
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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
−10.80024407239537620295547479286, −9.857008221633335858579318850007, −8.441706747389540052377284719339, −7.47045529874696752118658866819, −6.64377657157753930792782436350, −5.49687760086354509741908011774, −4.43729084390328053073320533530, −3.55990286837408158080034984225, −1.34327937123262441550404762117, −0.21715298591256696383692215825,
0.832517453706098061643229596280, 2.78443030897026439812842180275, 4.10920041099009648703834475426, 5.32506673782472841811693734839, 6.08121560649097302562107614414, 7.06698990441492103187646073696, 8.339426316432096571375833782004, 9.066591968480351164741448204695, 10.58092813932868110641253094847, 11.40157543829881045087923811113
