L(s) = 1 | + 2.66·5-s + (−0.654 − 2.56i)7-s + 3.98·11-s + (1.00 + 1.73i)13-s + (3.57 + 6.18i)17-s + (−4.01 + 6.96i)19-s + 0.887·23-s + 2.12·25-s + (1.35 − 2.33i)29-s + (0.614 − 1.06i)31-s + (−1.74 − 6.84i)35-s + (5.26 − 9.11i)37-s + (1.43 + 2.48i)41-s + (3.40 − 5.88i)43-s + (−6.06 − 10.5i)47-s + ⋯ |
L(s) = 1 | + 1.19·5-s + (−0.247 − 0.968i)7-s + 1.20·11-s + (0.277 + 0.480i)13-s + (0.866 + 1.50i)17-s + (−0.922 + 1.59i)19-s + 0.185·23-s + 0.424·25-s + (0.250 − 0.434i)29-s + (0.110 − 0.191i)31-s + (−0.295 − 1.15i)35-s + (0.865 − 1.49i)37-s + (0.224 + 0.388i)41-s + (0.518 − 0.898i)43-s + (−0.885 − 1.53i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.270942705\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.270942705\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.654 + 2.56i)T \) |
good | 5 | \( 1 - 2.66T + 5T^{2} \) |
| 11 | \( 1 - 3.98T + 11T^{2} \) |
| 13 | \( 1 + (-1.00 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.57 - 6.18i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.01 - 6.96i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 0.887T + 23T^{2} \) |
| 29 | \( 1 + (-1.35 + 2.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.614 + 1.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.26 + 9.11i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.43 - 2.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.40 + 5.88i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (6.06 + 10.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.38 - 4.13i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.79 + 8.29i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.74 - 8.22i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.49 - 9.51i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.62T + 71T^{2} \) |
| 73 | \( 1 + (-2.01 - 3.48i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.514 - 0.891i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.26 + 9.12i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.72 - 2.99i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.12 - 1.94i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.659456232416167609553202398419, −8.741193366313253290061477488521, −7.935395109197169182308916805179, −6.85319043417739506723181039085, −6.16433240573664239689202222062, −5.66731970647998609255891173263, −4.06435432708513158198680635733, −3.79096817694921616527826024847, −2.06325643097886445595363161466, −1.25424639885363004168167300201,
1.11886959395592055687292614105, 2.43695672433448656336623828636, 3.12993611803328312078568271484, 4.66019375377896697084915674348, 5.37630245141815151367842019297, 6.32183681824253498541343624461, 6.70322368050594149966610273770, 7.976436613290804653580993589311, 9.021817108011571045872332474212, 9.350214833991414868840613908006