| L(s) = 1 | + (−0.654 − 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (−0.415 + 0.909i)5-s + (0.142 + 0.989i)6-s + (−0.783 − 0.229i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (0.959 − 0.281i)10-s + (1.71 − 1.97i)11-s + (0.654 − 0.755i)12-s + (−2.29 + 0.672i)13-s + (0.339 + 0.742i)14-s + (0.841 − 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.649 + 4.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (−0.485 − 0.312i)3-s + (−0.0711 + 0.494i)4-s + (−0.185 + 0.406i)5-s + (0.0580 + 0.404i)6-s + (−0.295 − 0.0869i)7-s + (0.297 − 0.191i)8-s + (0.138 + 0.303i)9-s + (0.303 − 0.0890i)10-s + (0.516 − 0.596i)11-s + (0.189 − 0.218i)12-s + (−0.635 + 0.186i)13-s + (0.0906 + 0.198i)14-s + (0.217 − 0.139i)15-s + (−0.239 − 0.0704i)16-s + (0.157 + 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.440833 - 0.612465i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.440833 - 0.612465i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 3 | \( 1 + (0.841 + 0.540i)T \) |
| 5 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (-3.90 - 2.78i)T \) |
| good | 7 | \( 1 + (0.783 + 0.229i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-1.71 + 1.97i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.29 - 0.672i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.649 - 4.52i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-1.10 + 7.66i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (1.15 + 8.02i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-6.77 + 4.35i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (4.56 + 10.0i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-1.04 + 2.28i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (0.744 + 0.478i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 2.60T + 47T^{2} \) |
| 53 | \( 1 + (-0.800 - 0.235i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (1.85 - 0.543i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-4.71 + 3.03i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-3.11 - 3.59i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-0.135 - 0.156i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.54 + 10.7i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (15.4 - 4.54i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.283 - 0.620i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-8.86 - 5.69i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-5.54 + 12.1i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.26285060088267969264582908086, −9.475904286329317399271832829587, −8.583491883548753428387987148670, −7.53713974881107753461915596109, −6.80560844656748374816675773617, −5.86532821908851579832603717780, −4.54318324404368822432160102727, −3.41472857666253814409165211072, −2.21695447778916435333369879893, −0.55638703235674220805867817267,
1.28349711495378328946813933450, 3.20265456535939751871331885132, 4.65438311078548059837874160533, 5.24423818535695106250267529852, 6.43707475346755273270866349769, 7.14340994483583230659075584557, 8.151486116980207717364604670938, 9.065610896477304533358610292104, 9.877436267830805407275308282017, 10.38052972920236121416266562440
