| L(s) = 1 | + (−13.6 − 7.43i)3-s − 101. i·5-s − 49i·7-s + (132. + 203. i)9-s + 563.·11-s − 261.·13-s + (−752. + 1.38e3i)15-s + 2.05e3i·17-s + 978. i·19-s + (−364. + 671. i)21-s + 2.20e3·23-s − 7.12e3·25-s + (−297. − 3.77e3i)27-s + 5.60e3i·29-s + 4.73e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.878 − 0.477i)3-s − 1.81i·5-s − 0.377i·7-s + (0.544 + 0.838i)9-s + 1.40·11-s − 0.429·13-s + (−0.864 + 1.59i)15-s + 1.72i·17-s + 0.621i·19-s + (−0.180 + 0.332i)21-s + 0.870·23-s − 2.27·25-s + (−0.0785 − 0.996i)27-s + 1.23i·29-s + 0.885i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0262i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.322650397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.322650397\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (13.6 + 7.43i)T \) |
| 7 | \( 1 + 49iT \) |
| good | 5 | \( 1 + 101. iT - 3.12e3T^{2} \) |
| 11 | \( 1 - 563.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 261.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.05e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 978. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.20e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.60e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.73e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.08e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.77e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.23e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 6.70e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.48e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.14e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.92e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.24e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.32e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 6.22e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.75e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.78e5T + 8.58e9T^{2} \) |
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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
−10.87078691918031884041551082962, −9.754886398647111813323458122920, −8.792645040107864462412642393797, −7.971448017778425137076926236881, −6.73675282520712245557894288033, −5.78891082412581167382888549787, −4.79219661968927456374740983710, −3.97895371433500526996375904389, −1.47940643513587511635513940693, −1.09769483913268846352792351490,
0.46672645004082879607759416862, 2.43851918711408703697056401195, 3.52673117161196602511492831622, 4.73337141792357378100369119534, 6.06007306910362313444149984063, 6.72962212264507511620703247612, 7.46340761361516064820986187486, 9.379372464846573536989218881233, 9.722221768484467886271309703997, 10.99765080341818090486120894934
