| L(s) = 1 | + (0.939 + 0.342i)2-s + (0.515 + 2.92i)3-s + (0.766 + 0.642i)4-s + (2.88 − 2.42i)5-s + (−0.515 + 2.92i)6-s + (−1.44 + 2.50i)7-s + (0.500 + 0.866i)8-s + (−5.45 + 1.98i)9-s + (3.54 − 1.29i)10-s + (−0.5 − 0.866i)11-s + (−1.48 + 2.57i)12-s + (0.604 − 3.42i)13-s + (−2.21 + 1.85i)14-s + (8.57 + 7.19i)15-s + (0.173 + 0.984i)16-s + (0.739 + 0.269i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.297 + 1.68i)3-s + (0.383 + 0.321i)4-s + (1.29 − 1.08i)5-s + (−0.210 + 1.19i)6-s + (−0.546 + 0.947i)7-s + (0.176 + 0.306i)8-s + (−1.81 + 0.662i)9-s + (1.12 − 0.408i)10-s + (−0.150 − 0.261i)11-s + (−0.428 + 0.741i)12-s + (0.167 − 0.950i)13-s + (−0.592 + 0.497i)14-s + (2.21 + 1.85i)15-s + (0.0434 + 0.246i)16-s + (0.179 + 0.0652i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0391 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0391 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.69955 + 1.76737i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69955 + 1.76737i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-3.54 - 2.53i)T \) |
| good | 3 | \( 1 + (-0.515 - 2.92i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-2.88 + 2.42i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (1.44 - 2.50i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-0.604 + 3.42i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.739 - 0.269i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (5.66 + 4.75i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (6.98 - 2.54i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.13 + 5.42i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.89T + 37T^{2} \) |
| 41 | \( 1 + (0.0504 + 0.286i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (1.13 - 0.955i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (2.19 - 0.798i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (6.14 + 5.15i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-13.1 - 4.79i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.549 - 0.460i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-11.5 + 4.18i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (6.69 - 5.61i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.661 - 3.75i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (2.12 + 12.0i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (7.35 - 12.7i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.82 + 16.0i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-12.7 - 4.63i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.41581163455458768443750845034, −10.11866724555690407444141270997, −9.782139236263386517128253644858, −8.865944344599434332942825753736, −8.156143099045374972150746920962, −5.91152773631327707772733649228, −5.67115234570096594795725584443, −4.78143233164154447101612686290, −3.58136431875814287274402193609, −2.43525087657189531085710858958,
1.54218085409187226422272940749, 2.47324132322619956548521178282, 3.58626601706636764326961042021, 5.55430065821158159140318301469, 6.50203083165592383002307095032, 6.93415542831837355061077433757, 7.71295296584078288647758993785, 9.390542848607461209655429942224, 10.10720888308558169296762652894, 11.22068117227258340638361582049
