| L(s) = 1 | + (1.38 + 0.289i)2-s + (0.504 − 1.65i)3-s + (1.83 + 0.802i)4-s + (1.79 + 0.316i)5-s + (1.17 − 2.14i)6-s + (−1.94 + 1.63i)7-s + (2.30 + 1.64i)8-s + (−2.49 − 1.67i)9-s + (2.39 + 0.958i)10-s + (−5.63 + 0.992i)11-s + (2.25 − 2.63i)12-s + (−0.682 − 1.87i)13-s + (−3.17 + 1.69i)14-s + (1.43 − 2.81i)15-s + (2.71 + 2.94i)16-s + (2.21 − 3.84i)17-s + ⋯ |
| L(s) = 1 | + (0.978 + 0.205i)2-s + (0.291 − 0.956i)3-s + (0.915 + 0.401i)4-s + (0.802 + 0.141i)5-s + (0.481 − 0.876i)6-s + (−0.736 + 0.617i)7-s + (0.814 + 0.580i)8-s + (−0.830 − 0.557i)9-s + (0.756 + 0.303i)10-s + (−1.69 + 0.299i)11-s + (0.650 − 0.759i)12-s + (−0.189 − 0.520i)13-s + (−0.847 + 0.453i)14-s + (0.369 − 0.726i)15-s + (0.677 + 0.735i)16-s + (0.538 − 0.932i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.22958 - 0.253410i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.22958 - 0.253410i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 - 0.289i)T \) |
| 3 | \( 1 + (-0.504 + 1.65i)T \) |
| good | 5 | \( 1 + (-1.79 - 0.316i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (1.94 - 1.63i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (5.63 - 0.992i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (0.682 + 1.87i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.21 + 3.84i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.29 - 0.746i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.26 - 5.26i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.689 + 1.89i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (4.79 + 4.01i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-0.718 - 0.414i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.05 + 0.382i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (4.19 - 0.740i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.51 + 5.47i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 11.7iT - 53T^{2} \) |
| 59 | \( 1 + (-12.4 - 2.20i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.29 - 5.12i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (4.70 + 12.9i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.72 + 4.71i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.95 + 5.12i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.43 + 1.61i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.707 - 1.94i)T + (-63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (0.113 + 0.195i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.744 - 4.21i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.67362363389735349745034585277, −11.73185601718501102859493343550, −10.47415470010465255291374307670, −9.323624444838239027169485523117, −7.86142256538189400843734566510, −7.14496357912458616226591581503, −5.86517089740067850257504356889, −5.34230304775787449466653706429, −3.07566713702454242113180741755, −2.32792053566823714085089796890,
2.48639407098708619344124849003, 3.61946983054830335285608530390, 4.93284981381330737852041812433, 5.74360062756819240036840601056, 7.03174834309606892830025103650, 8.503382454385717308781242037625, 9.904153845832670725936127458839, 10.38377159245910326890203881519, 11.16680485119248414605625662833, 12.85683756228856919886369851253
