| L(s) = 1 | + (−2.93 + 0.640i)3-s − 6.71i·5-s − 3.02·7-s + (8.17 − 3.75i)9-s + 0.160·13-s + (4.30 + 19.6i)15-s − 24.4i·17-s − 0.622·19-s + (8.87 − 1.93i)21-s − 6.09i·23-s − 20.0·25-s + (−21.5 + 16.2i)27-s − 52.2i·29-s + 44.6·31-s + 20.3i·35-s + ⋯ |
| L(s) = 1 | + (−0.976 + 0.213i)3-s − 1.34i·5-s − 0.432·7-s + (0.908 − 0.417i)9-s + 0.0123·13-s + (0.286 + 1.31i)15-s − 1.43i·17-s − 0.0327·19-s + (0.422 − 0.0923i)21-s − 0.264i·23-s − 0.802·25-s + (−0.798 + 0.601i)27-s − 1.80i·29-s + 1.43·31-s + 0.580i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.213i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.976 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7943496068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7943496068\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.93 - 0.640i)T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 6.71iT - 25T^{2} \) |
| 7 | \( 1 + 3.02T + 49T^{2} \) |
| 13 | \( 1 - 0.160T + 169T^{2} \) |
| 17 | \( 1 + 24.4iT - 289T^{2} \) |
| 19 | \( 1 + 0.622T + 361T^{2} \) |
| 23 | \( 1 + 6.09iT - 529T^{2} \) |
| 29 | \( 1 + 52.2iT - 841T^{2} \) |
| 31 | \( 1 - 44.6T + 961T^{2} \) |
| 37 | \( 1 - 30.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 1.89iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 79.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 14.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 47.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 21.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 61.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 56.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 70.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 98.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 103. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 165. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 97.8T + 9.40e3T^{2} \) |
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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
−9.283326639864702681499037502306, −8.160083983988184361433348057630, −7.31642194432079449783129194978, −6.26423197170790160745984456932, −5.64642065825623615296158703307, −4.63519534233031466329985629412, −4.29330825789342891366101611390, −2.71135041857418259963982798383, −1.09847671842879091814762306487, −0.30621008459737330908432906614,
1.37237143533918163301354302618, 2.69265175816783744288199201171, 3.68715337761793919810309553352, 4.72490988568171062687204582745, 5.97011069357182353602103515428, 6.31814892949993583168601949234, 7.12495044454217869600426105601, 7.81566997058876441619211723290, 8.972805214494642786295493918063, 10.14072747257849733314169595636
