L(s) = 1 | + (−1.73 − 0.0755i)3-s + 0.967·5-s + (−1.11 + 2.39i)7-s + (2.98 + 0.261i)9-s + 0.728·11-s + (1.81 + 3.13i)13-s + (−1.67 − 0.0731i)15-s + (3.49 + 6.06i)17-s + (−0.348 + 0.602i)19-s + (2.11 − 4.06i)21-s + 6.43·23-s − 4.06·25-s + (−5.15 − 0.678i)27-s + (3.34 − 5.79i)29-s + (−4.58 + 7.93i)31-s + ⋯ |
L(s) = 1 | + (−0.999 − 0.0436i)3-s + 0.432·5-s + (−0.421 + 0.906i)7-s + (0.996 + 0.0871i)9-s + 0.219·11-s + (0.502 + 0.869i)13-s + (−0.432 − 0.0188i)15-s + (0.848 + 1.47i)17-s + (−0.0798 + 0.138i)19-s + (0.460 − 0.887i)21-s + 1.34·23-s − 0.812·25-s + (−0.991 − 0.130i)27-s + (0.621 − 1.07i)29-s + (−0.823 + 1.42i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.849046 + 0.429899i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.849046 + 0.429899i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.73 + 0.0755i)T \) |
| 7 | \( 1 + (1.11 - 2.39i)T \) |
good | 5 | \( 1 - 0.967T + 5T^{2} \) |
| 11 | \( 1 - 0.728T + 11T^{2} \) |
| 13 | \( 1 + (-1.81 - 3.13i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.49 - 6.06i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.348 - 0.602i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 6.43T + 23T^{2} \) |
| 29 | \( 1 + (-3.34 + 5.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.58 - 7.93i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.854 + 1.48i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.62 + 6.27i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.348 - 0.602i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.83 + 6.63i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.05 - 3.56i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.38 + 4.13i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.46 + 4.26i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.91 + 5.04i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.304T + 71T^{2} \) |
| 73 | \( 1 + (-5.33 - 9.24i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.61 - 2.80i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.618 + 1.07i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (5.78 - 10.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.32 + 2.30i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.21254050102996806606519531564, −11.30650393960380466688253081234, −10.33868483159964528251304927286, −9.441856767557045195707026080370, −8.402146173457391672636717695926, −6.86196979583613177258245789628, −6.07850862716383830425586936980, −5.24654755940994269454245048943, −3.74077476252067360009539968370, −1.74720809653009538082668078772,
0.947372097348023897407639785286, 3.31132625728542440156023607267, 4.77255696217525335687639721039, 5.77520833437970793139994376194, 6.82890688452275551310932509258, 7.68090796045734067604852681986, 9.355796243924277447481477088888, 10.08756752406077443544936129710, 10.94969933936571337010668397335, 11.76263744332934466991528230743