L(s) = 1 | + (0.5 + 0.866i)3-s + (1 − 1.73i)5-s + (−0.499 + 0.866i)9-s + (2 + 3.46i)11-s − 2·13-s + 1.99·15-s + (3 + 5.19i)17-s + (2 − 3.46i)19-s + (0.500 + 0.866i)25-s − 0.999·27-s − 2·29-s + (−1.99 + 3.46i)33-s + (−3 + 5.19i)37-s + (−1 − 1.73i)39-s + 2·41-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.447 − 0.774i)5-s + (−0.166 + 0.288i)9-s + (0.603 + 1.04i)11-s − 0.554·13-s + 0.516·15-s + (0.727 + 1.26i)17-s + (0.458 − 0.794i)19-s + (0.100 + 0.173i)25-s − 0.192·27-s − 0.371·29-s + (−0.348 + 0.603i)33-s + (−0.493 + 0.854i)37-s + (−0.160 − 0.277i)39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.166858736\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.166858736\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-3 - 5.19i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.141586222502462082609396679405, −8.525585438477936160443132809337, −7.60716961882873176513262596023, −6.85300057974494424073083374816, −5.79227885712163690276153584226, −5.04949071521281981685592426534, −4.36813759831279827187110938651, −3.45972651216623040377621903417, −2.24281320202841076040231901051, −1.25472338647344124511488483847,
0.78354883221851688463399060692, 2.10471627684087977317770597999, 3.02200297793579465275070352352, 3.69485898387662226937709894375, 5.07172018458693842342994873903, 5.90441797912309754295200006364, 6.54951996829193710377981229934, 7.39183450447668903519168005615, 7.912485992073285104694939895389, 8.986110837635124187554553854430