L(s) = 1 | + (−1.21 − 2.10i)2-s − 0.753·3-s + (−1.95 + 3.39i)4-s + (0.170 − 0.295i)5-s + (0.916 + 1.58i)6-s + 4.65·8-s − 2.43·9-s − 0.830·10-s − 2.43·11-s + (1.47 − 2.55i)12-s + (−2.50 − 2.59i)13-s + (−0.128 + 0.222i)15-s + (−1.74 − 3.02i)16-s + (−0.974 + 1.68i)17-s + (2.95 + 5.12i)18-s + 6.29·19-s + ⋯ |
L(s) = 1 | + (−0.859 − 1.48i)2-s − 0.435·3-s + (−0.978 + 1.69i)4-s + (0.0763 − 0.132i)5-s + (0.374 + 0.647i)6-s + 1.64·8-s − 0.810·9-s − 0.262·10-s − 0.733·11-s + (0.425 − 0.737i)12-s + (−0.693 − 0.720i)13-s + (−0.0332 + 0.0575i)15-s + (−0.437 − 0.757i)16-s + (−0.236 + 0.409i)17-s + (0.697 + 1.20i)18-s + 1.44·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.461474 - 0.0814343i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.461474 - 0.0814343i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (2.50 + 2.59i)T \) |
good | 2 | \( 1 + (1.21 + 2.10i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + 0.753T + 3T^{2} \) |
| 5 | \( 1 + (-0.170 + 0.295i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 2.43T + 11T^{2} \) |
| 17 | \( 1 + (0.974 - 1.68i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 + (-1.84 - 3.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.22 - 3.84i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.987 + 1.71i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.81 - 8.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.26 + 10.8i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.20 - 7.28i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.50 - 7.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.746 + 1.29i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.313 + 0.542i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 + 5.59T + 67T^{2} \) |
| 71 | \( 1 + (4.74 + 8.22i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.95 - 10.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.23 - 3.87i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + (-6.22 - 10.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.13 - 8.90i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.73550586385557232755902777308, −9.723173792031318309872892210295, −9.175828095089196383112123791038, −8.129950037824926757744425502170, −7.42211906390402054450233642412, −5.77225416052931937572521627278, −4.92871286233877608047780341961, −3.34831203611429770078683368418, −2.62962589377810023951117722675, −1.07127041897972966231277231557,
0.43554863557981812431947576478, 2.67951226208558297893152438853, 4.68577219376450336121145293966, 5.47483197936505955306016065268, 6.26236972055888016217106639848, 7.17824343932956534213094136016, 7.82603199682362159638732057044, 8.815036782479067317439091884169, 9.483113236605014283306307954492, 10.33748236179586269192298144554