L(s) = 1 | + (−0.341 + 1.69i)3-s + 0.526·5-s + (2.43 − 1.02i)7-s + (−2.76 − 1.16i)9-s + 4.61·11-s + (0.244 + 0.423i)13-s + (−0.179 + 0.893i)15-s + (2.75 + 4.77i)17-s + (1.83 − 3.18i)19-s + (0.904 + 4.49i)21-s − 0.0539·23-s − 4.72·25-s + (2.91 − 4.30i)27-s + (−3.28 + 5.68i)29-s + (−3.03 + 5.26i)31-s + ⋯ |
L(s) = 1 | + (−0.197 + 0.980i)3-s + 0.235·5-s + (0.922 − 0.386i)7-s + (−0.922 − 0.386i)9-s + 1.39·11-s + (0.0678 + 0.117i)13-s + (−0.0464 + 0.230i)15-s + (0.668 + 1.15i)17-s + (0.421 − 0.730i)19-s + (0.197 + 0.980i)21-s − 0.0112·23-s − 0.944·25-s + (0.561 − 0.827i)27-s + (−0.609 + 1.05i)29-s + (−0.545 + 0.945i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41501 + 0.715389i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41501 + 0.715389i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.341 - 1.69i)T \) |
| 7 | \( 1 + (-2.43 + 1.02i)T \) |
good | 5 | \( 1 - 0.526T + 5T^{2} \) |
| 11 | \( 1 - 4.61T + 11T^{2} \) |
| 13 | \( 1 + (-0.244 - 0.423i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.75 - 4.77i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.83 + 3.18i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 0.0539T + 23T^{2} \) |
| 29 | \( 1 + (3.28 - 5.68i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.03 - 5.26i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.223 + 0.387i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.52 - 4.36i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.84 + 4.93i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.59 - 7.96i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.37 + 7.57i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.31 + 5.74i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.232 - 0.403i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.59 - 4.49i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.76T + 71T^{2} \) |
| 73 | \( 1 + (5.23 + 9.07i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.18 + 14.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.49 + 7.78i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (7.05 - 12.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.22 + 9.04i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.04094865224842561970373375196, −10.22341040583436892025669114809, −9.282681184215532341123264653769, −8.630724672660154862552052990931, −7.46659271068569345428886315072, −6.26853769487439721236942643740, −5.32072959736495867300669139320, −4.29091438846897907825908674306, −3.46758491578798000542711925492, −1.52849624318242406249539035115,
1.21460519973619092970711817697, 2.36968825754051751187730441321, 3.98802084907591521889385300040, 5.45325534494448147704357868536, 6.04525190203253694700172631995, 7.29279342088072544987027699877, 7.88788368929033202230037570823, 8.960397646362298184144992677355, 9.759662846625592551033798324745, 11.16662096704495971067276262513