L(s) = 1 | + (1.26 − 0.637i)2-s + (0.988 + 0.149i)3-s + (1.18 − 1.60i)4-s + (1.46 − 0.575i)5-s + (1.34 − 0.442i)6-s + (1.59 + 2.11i)7-s + (0.473 − 2.78i)8-s + (0.955 + 0.294i)9-s + (1.48 − 1.65i)10-s + (−0.356 − 1.15i)11-s + (1.41 − 1.41i)12-s + (−4.55 − 1.04i)13-s + (3.35 + 1.65i)14-s + (1.53 − 0.350i)15-s + (−1.17 − 3.82i)16-s + (−3.40 + 4.98i)17-s + ⋯ |
L(s) = 1 | + (0.892 − 0.450i)2-s + (0.570 + 0.0860i)3-s + (0.593 − 0.804i)4-s + (0.655 − 0.257i)5-s + (0.548 − 0.180i)6-s + (0.601 + 0.799i)7-s + (0.167 − 0.985i)8-s + (0.318 + 0.0982i)9-s + (0.469 − 0.524i)10-s + (−0.107 − 0.348i)11-s + (0.408 − 0.408i)12-s + (−1.26 − 0.288i)13-s + (0.896 + 0.442i)14-s + (0.396 − 0.0904i)15-s + (−0.294 − 0.955i)16-s + (−0.824 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 + 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.02994 - 1.16652i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.02994 - 1.16652i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.26 + 0.637i)T \) |
| 3 | \( 1 + (-0.988 - 0.149i)T \) |
| 7 | \( 1 + (-1.59 - 2.11i)T \) |
good | 5 | \( 1 + (-1.46 + 0.575i)T + (3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (0.356 + 1.15i)T + (-9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (4.55 + 1.04i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (3.40 - 4.98i)T + (-6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-1.30 + 2.26i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.84 - 5.63i)T + (-8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (1.17 - 0.563i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (1.77 + 3.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.292 + 3.90i)T + (-36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (3.78 + 3.01i)T + (9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-3.36 + 2.68i)T + (9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (0.449 + 0.417i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (0.651 + 8.69i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (0.100 - 0.256i)T + (-43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (7.65 + 0.573i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (-9.08 + 5.24i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.56 - 11.5i)T + (-44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-7.52 - 8.10i)T + (-5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-6.37 - 3.68i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.54 - 6.75i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (3.13 - 10.1i)T + (-73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 8.77iT - 97T^{2} \) |
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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
−10.76515838843366847732698395485, −9.654278267563711459153616048160, −9.126648857461954227737505286956, −7.942273259336602932155925970432, −6.86465363834819875892421879659, −5.57740664514863019927183851214, −5.13561870430585816834267354762, −3.86103237356421612272080827316, −2.58378473286767668543846089476, −1.77690337980246975440406581093,
2.03843263809148630872803243756, 2.98343441486222983869097688841, 4.46579590180324557318765275941, 4.96157272195518599796787154874, 6.40814051975885106321393217684, 7.18833863685630072619064105434, 7.78510196519698972397547532518, 8.959605732540462118389572159428, 9.953578291569187221626874075518, 10.84424952620552041575118741078