L(s) = 1 | + (−1.10 + 1.33i)3-s + 5-s − 2.36i·7-s + (−0.540 − 2.95i)9-s − 0.927·11-s − 1.43i·13-s + (−1.10 + 1.33i)15-s − 5.72i·17-s − 1.85·19-s + (3.14 + 2.62i)21-s + 7.68i·23-s + 25-s + (4.52 + 2.55i)27-s − 7.94i·29-s + 0.732i·31-s + ⋯ |
L(s) = 1 | + (−0.640 + 0.768i)3-s + 0.447·5-s − 0.893i·7-s + (−0.180 − 0.983i)9-s − 0.279·11-s − 0.397i·13-s + (−0.286 + 0.343i)15-s − 1.38i·17-s − 0.426·19-s + (0.686 + 0.572i)21-s + 1.60i·23-s + 0.200·25-s + (0.870 + 0.491i)27-s − 1.47i·29-s + 0.131i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2483940194\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2483940194\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.10 - 1.33i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (7.02 + 4.19i)T \) |
good | 7 | \( 1 + 2.36iT - 7T^{2} \) |
| 11 | \( 1 + 0.927T + 11T^{2} \) |
| 13 | \( 1 + 1.43iT - 13T^{2} \) |
| 17 | \( 1 + 5.72iT - 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 - 7.68iT - 23T^{2} \) |
| 29 | \( 1 + 7.94iT - 29T^{2} \) |
| 31 | \( 1 - 0.732iT - 31T^{2} \) |
| 37 | \( 1 + 1.54T + 37T^{2} \) |
| 41 | \( 1 + 6.32T + 41T^{2} \) |
| 43 | \( 1 - 5.59iT - 43T^{2} \) |
| 47 | \( 1 + 9.43iT - 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 6.47iT - 59T^{2} \) |
| 61 | \( 1 - 13.3iT - 61T^{2} \) |
| 71 | \( 1 + 5.24iT - 71T^{2} \) |
| 73 | \( 1 + 3.58T + 73T^{2} \) |
| 79 | \( 1 - 12.6iT - 79T^{2} \) |
| 83 | \( 1 - 4.64iT - 83T^{2} \) |
| 89 | \( 1 + 1.75iT - 89T^{2} \) |
| 97 | \( 1 - 12.2iT - 97T^{2} \) |
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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
−8.057538982467282579219559418658, −7.30414024132988919937228891022, −6.60874058598281741270070101837, −5.73200793113645704542200671975, −5.15564811153461534537332166833, −4.38808625396567557845691431255, −3.58572131280934609392944548926, −2.69684346129017836521781187509, −1.24652186911167992448947421168, −0.07870970497146588275280039362,
1.53541426796723910309075196842, 2.15926575140209155459827897015, 3.14316018344458816199732637062, 4.48774533181444180278185075410, 5.16958953391251338175489112749, 6.01003203879456885175307668419, 6.40696231220909019682835094892, 7.13051966348892125946583652396, 8.203481578371148931515338615714, 8.556286632457795578572394859114