| L(s) = 1 | + (1.08 − 0.910i)2-s + (0.841 − 0.540i)3-s + (0.343 − 1.97i)4-s + (1.39 + 3.05i)5-s + (0.418 − 1.35i)6-s + (0.483 − 0.142i)7-s + (−1.42 − 2.44i)8-s + (0.415 − 0.909i)9-s + (4.29 + 2.03i)10-s + (−0.300 + 0.260i)11-s + (−0.776 − 1.84i)12-s + (0.849 − 2.89i)13-s + (0.394 − 0.593i)14-s + (2.82 + 1.81i)15-s + (−3.76 − 1.35i)16-s + (−2.38 − 0.343i)17-s + ⋯ |
| L(s) = 1 | + (0.765 − 0.643i)2-s + (0.485 − 0.312i)3-s + (0.171 − 0.985i)4-s + (0.624 + 1.36i)5-s + (0.170 − 0.551i)6-s + (0.182 − 0.0536i)7-s + (−0.502 − 0.864i)8-s + (0.138 − 0.303i)9-s + (1.35 + 0.645i)10-s + (−0.0907 + 0.0786i)11-s + (−0.224 − 0.532i)12-s + (0.235 − 0.802i)13-s + (0.105 − 0.158i)14-s + (0.730 + 0.469i)15-s + (−0.940 − 0.338i)16-s + (−0.579 − 0.0832i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.50119 - 1.28209i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.50119 - 1.28209i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.08 + 0.910i)T \) |
| 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (-1.90 - 4.40i)T \) |
| good | 5 | \( 1 + (-1.39 - 3.05i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.483 + 0.142i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (0.300 - 0.260i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.849 + 2.89i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (2.38 + 0.343i)T + (16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-8.08 + 1.16i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (4.03 + 0.580i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (0.00139 - 0.00217i)T + (-12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (2.74 - 6.00i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (1.88 + 4.12i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (2.65 + 4.13i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 6.52iT - 47T^{2} \) |
| 53 | \( 1 + (9.03 - 2.65i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (9.69 + 2.84i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (4.54 + 2.92i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (0.248 + 0.215i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-8.31 - 7.20i)T + (10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.470 + 3.27i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (12.7 + 3.72i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-10.3 - 4.73i)T + (54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-6.35 - 9.89i)T + (-36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-1.54 + 0.705i)T + (63.5 - 73.3i)T^{2} \) |
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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
−10.84788717527543929641623434029, −9.891529103709947191796072198138, −9.295630373637065864593094713327, −7.71348849369402223599436984566, −6.92441729683878773360153792103, −5.99715993808935471946965871465, −5.04440649402532913911267315240, −3.40162626575428403840690510284, −2.90315301802938250135684848906, −1.61823958139332072508573370799,
1.82341318413579238875404713559, 3.35688099014597593673494523293, 4.57969253151830695956958712142, 5.13978541807766965194046737942, 6.14547573725776959835044655084, 7.33243495463037040668203985050, 8.332886365765236069144153318769, 9.008259028128916120512905941037, 9.658753962608588671110143939468, 11.12274918816543257279389531904
