L(s) = 1 | + (−0.582 + 2.17i)2-s + (−1.60 + 0.644i)3-s + (−2.64 − 1.52i)4-s + (−2.21 + 0.337i)5-s + (−0.465 − 3.86i)6-s + (1.68 − 1.68i)8-s + (2.16 − 2.07i)9-s + (0.552 − 4.99i)10-s + (−3.88 − 2.24i)11-s + (5.24 + 0.750i)12-s + (1.08 + 1.08i)13-s + (3.33 − 1.96i)15-s + (−0.381 − 0.660i)16-s + (−2.04 + 0.548i)17-s + (3.24 + 5.91i)18-s + (3.66 − 2.11i)19-s + ⋯ |
L(s) = 1 | + (−0.411 + 1.53i)2-s + (−0.928 + 0.372i)3-s + (−1.32 − 0.764i)4-s + (−0.988 + 0.151i)5-s + (−0.189 − 1.57i)6-s + (0.595 − 0.595i)8-s + (0.722 − 0.691i)9-s + (0.174 − 1.58i)10-s + (−1.17 − 0.676i)11-s + (1.51 + 0.216i)12-s + (0.300 + 0.300i)13-s + (0.861 − 0.508i)15-s + (−0.0952 − 0.165i)16-s + (−0.496 + 0.133i)17-s + (0.764 + 1.39i)18-s + (0.839 − 0.484i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.361534 + 0.235217i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.361534 + 0.235217i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.60 - 0.644i)T \) |
| 5 | \( 1 + (2.21 - 0.337i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.582 - 2.17i)T + (-1.73 - i)T^{2} \) |
| 11 | \( 1 + (3.88 + 2.24i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.08 - 1.08i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.04 - 0.548i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.66 + 2.11i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.13 - 0.840i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 1.69T + 29T^{2} \) |
| 31 | \( 1 + (0.530 - 0.918i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.75 + 1.54i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 5.84iT - 41T^{2} \) |
| 43 | \( 1 + (-2.00 - 2.00i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.36 + 5.10i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.23 - 8.34i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-2.35 + 4.07i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.88 + 6.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.152 + 0.569i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 4.66iT - 71T^{2} \) |
| 73 | \( 1 + (-4.22 + 1.13i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.78 + 3.33i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.0 + 11.0i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.75 - 3.04i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.60 + 5.60i)T - 97iT^{2} \) |
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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
−10.60462255263311387666782932212, −9.410235726089418821042836460039, −8.640989489381881597147799680300, −7.72170709231177693595252820378, −7.08250916950702899656197429093, −6.23802282975820351871979542151, −5.30592262613123396291595269746, −4.64793988526283539967548949175, −3.29900615081890410054712635912, −0.43026266146903352231301412502,
0.834686281752472580597351168540, 2.23660512032413777930990207724, 3.49107160951539990228609358230, 4.56726656224884262543693818959, 5.44773762675456092154360242627, 6.95299638433695087063722381689, 7.73362062181687901779336038702, 8.642959118032416178562943545526, 9.746157475612171310848922644232, 10.58628850971194266749631710151