L(s) = 1 | + (0.943 − 1.05i)2-s + (−0.219 − 1.98i)4-s + (−2.75 + 1.59i)5-s + (0.944 + 2.47i)7-s + (−2.30 − 1.64i)8-s + (−0.924 + 4.40i)10-s + (−1.24 + 2.16i)11-s + 6.61·13-s + (3.49 + 1.33i)14-s + (−3.90 + 0.874i)16-s + (2.67 + 1.54i)17-s + (−4.40 + 2.54i)19-s + (3.77 + 5.13i)20-s + (1.10 + 3.35i)22-s + (2.53 + 4.39i)23-s + ⋯ |
L(s) = 1 | + (0.667 − 0.744i)2-s + (−0.109 − 0.993i)4-s + (−1.23 + 0.712i)5-s + (0.356 + 0.934i)7-s + (−0.813 − 0.581i)8-s + (−0.292 + 1.39i)10-s + (−0.376 + 0.651i)11-s + 1.83·13-s + (0.933 + 0.357i)14-s + (−0.975 + 0.218i)16-s + (0.649 + 0.374i)17-s + (−1.01 + 0.583i)19-s + (0.843 + 1.14i)20-s + (0.234 + 0.715i)22-s + (0.529 + 0.916i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58380 + 0.330714i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58380 + 0.330714i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.943 + 1.05i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.944 - 2.47i)T \) |
good | 5 | \( 1 + (2.75 - 1.59i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.24 - 2.16i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6.61T + 13T^{2} \) |
| 17 | \( 1 + (-2.67 - 1.54i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.40 - 2.54i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.53 - 4.39i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.59iT - 29T^{2} \) |
| 31 | \( 1 + (-7.31 - 4.22i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.357 + 0.619i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 12.2iT - 41T^{2} \) |
| 43 | \( 1 + 3.72iT - 43T^{2} \) |
| 47 | \( 1 + (2.51 + 4.35i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.21 + 4.16i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.36 + 9.29i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.997 - 1.72i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.00277 - 0.00160i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + (-0.957 + 1.65i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.11 - 3.52i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.93T + 83T^{2} \) |
| 89 | \( 1 + (0.482 - 0.278i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 0.127T + 97T^{2} \) |
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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
−10.70463947292310105091301887229, −9.836105209387387429232784355973, −8.580349534027448928756391803130, −8.009830955857664934611293315687, −6.66980999280991523898476833333, −5.89671885399433207675708592442, −4.76168095569527493602102606500, −3.75197233879923120487325631013, −3.02559134148746334347997301451, −1.60000539836115413610401569525,
0.72076941095656536449802249340, 3.16042254172701560613531385430, 4.10568684057416711842857329141, 4.63003680635079069344627890383, 5.84910648506930192231398216372, 6.80527125915306060710475466804, 7.77573935195358623020977462338, 8.365850977337810791062671326878, 8.883823061142363756376665100735, 10.62425876618894154045701965887