| L(s) = 1 | + (0.888 − 0.458i)2-s + (−0.818 + 1.52i)3-s + (0.580 − 0.814i)4-s + (−0.448 − 1.84i)5-s + (−0.0283 + 1.73i)6-s + (−1.08 − 3.13i)7-s + (0.142 − 0.989i)8-s + (−1.65 − 2.49i)9-s + (−1.24 − 1.43i)10-s + (−0.0584 − 1.22i)11-s + (0.768 + 1.55i)12-s + (0.00414 − 0.0119i)13-s + (−2.40 − 2.29i)14-s + (3.18 + 0.829i)15-s + (−0.327 − 0.945i)16-s + (−1.33 + 2.92i)17-s + ⋯ |
| L(s) = 1 | + (0.628 − 0.324i)2-s + (−0.472 + 0.881i)3-s + (0.290 − 0.407i)4-s + (−0.200 − 0.827i)5-s + (−0.0115 + 0.707i)6-s + (−0.410 − 1.18i)7-s + (0.0503 − 0.349i)8-s + (−0.553 − 0.833i)9-s + (−0.394 − 0.454i)10-s + (−0.0176 − 0.370i)11-s + (0.221 + 0.448i)12-s + (0.00114 − 0.00332i)13-s + (−0.642 − 0.612i)14-s + (0.823 + 0.214i)15-s + (−0.0817 − 0.236i)16-s + (−0.323 + 0.708i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0302 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0302 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.932729 - 0.961378i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.932729 - 0.961378i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.888 + 0.458i)T \) |
| 3 | \( 1 + (0.818 - 1.52i)T \) |
| 23 | \( 1 + (-3.64 + 3.11i)T \) |
| good | 5 | \( 1 + (0.448 + 1.84i)T + (-4.44 + 2.29i)T^{2} \) |
| 7 | \( 1 + (1.08 + 3.13i)T + (-5.50 + 4.32i)T^{2} \) |
| 11 | \( 1 + (0.0584 + 1.22i)T + (-10.9 + 1.04i)T^{2} \) |
| 13 | \( 1 + (-0.00414 + 0.0119i)T + (-10.2 - 8.03i)T^{2} \) |
| 17 | \( 1 + (1.33 - 2.92i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (1.95 + 4.28i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-1.34 - 1.89i)T + (-9.48 + 27.4i)T^{2} \) |
| 31 | \( 1 + (0.618 - 0.247i)T + (22.4 - 21.3i)T^{2} \) |
| 37 | \( 1 + (-8.14 + 2.39i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-2.08 - 8.57i)T + (-36.4 + 18.7i)T^{2} \) |
| 43 | \( 1 + (8.50 + 3.40i)T + (31.1 + 29.6i)T^{2} \) |
| 47 | \( 1 + (-1.21 + 2.11i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.34 - 5.00i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (3.28 - 9.49i)T + (-46.3 - 36.4i)T^{2} \) |
| 61 | \( 1 + (-1.83 - 1.44i)T + (14.3 + 59.2i)T^{2} \) |
| 67 | \( 1 + (0.0474 - 0.995i)T + (-66.6 - 6.36i)T^{2} \) |
| 71 | \( 1 + (-2.29 - 1.47i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (0.782 + 1.71i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (3.42 - 0.660i)T + (73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (-3.06 + 12.6i)T + (-73.7 - 38.0i)T^{2} \) |
| 89 | \( 1 + (0.0118 + 0.0821i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-8.25 + 7.87i)T + (4.61 - 96.8i)T^{2} \) |
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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
−10.88797903080951185373640099867, −10.41765521462874078979775952818, −9.336271436250153897785195717158, −8.468045177065285211630310808705, −6.95283431288673474079789696037, −6.04296044754353889819269186949, −4.74034769566862210450686572466, −4.29544002118908684224982242181, −3.12222450261331809351606796166, −0.75316938697436088282004659876,
2.20045076838486781522970333141, 3.21135190563121517940243891258, 4.92482493351067476241232598138, 5.91806702926223541714348090048, 6.62865944504550648879238078618, 7.44755448232585212206334336110, 8.424279735672369598150596439427, 9.641821240406512658864975655262, 10.96507396133018164862806752029, 11.60532343444728808374542996811
