L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 3·5-s + (0.499 + 0.866i)6-s + (0.5 + 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.5 + 2.59i)10-s + (−2 + 3.46i)11-s − 0.999·12-s + (3.5 + 0.866i)13-s − 0.999·14-s + (1.5 − 2.59i)15-s + (−0.5 + 0.866i)16-s + (2.5 + 4.33i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + 1.34·5-s + (0.204 + 0.353i)6-s + (0.188 + 0.327i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.474 + 0.821i)10-s + (−0.603 + 1.04i)11-s − 0.288·12-s + (0.970 + 0.240i)13-s − 0.267·14-s + (0.387 − 0.670i)15-s + (−0.125 + 0.216i)16-s + (0.606 + 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.60565 + 0.441288i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.60565 + 0.441288i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-3.5 - 0.866i)T \) |
good | 5 | \( 1 - 3T + 5T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.5 - 6.06i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 7T + 53T^{2} \) |
| 59 | \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6 + 10.3i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9 + 15.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.41030222580722124453489354693, −10.03446004932540000580444650919, −8.865907095545422819853026582045, −8.382321443687000781853379714171, −7.19512410512835180601112530032, −6.32151837875218101606599151865, −5.66005246864061023392537503436, −4.47035994590382656832482485186, −2.55518251274908605510094966011, −1.52907911793429289417876431628,
1.31391052449536356878880577333, 2.74247361109148928325391222304, 3.66893432987448413897029189361, 5.19928598269149437855311638599, 5.86304369308204668490255619496, 7.27980534391095969980245568447, 8.479032050064001711359886192138, 9.014479191310767591713347856317, 10.03784542072222927536970703625, 10.53343115532845636195354206352