| L(s) = 1 | + (−0.877 + 0.479i)2-s + (−0.997 + 0.0713i)3-s + (0.540 − 0.841i)4-s + (2.11 + 0.714i)5-s + (0.841 − 0.540i)6-s + (0.271 − 0.726i)7-s + (−0.0713 + 0.997i)8-s + (0.989 − 0.142i)9-s + (−2.20 + 0.388i)10-s + (−1.66 − 5.65i)11-s + (−0.479 + 0.877i)12-s + (3.64 − 1.36i)13-s + (0.110 + 0.767i)14-s + (−2.16 − 0.561i)15-s + (−0.415 − 0.909i)16-s + (−3.47 − 0.756i)17-s + ⋯ |
| L(s) = 1 | + (−0.620 + 0.338i)2-s + (−0.575 + 0.0411i)3-s + (0.270 − 0.420i)4-s + (0.947 + 0.319i)5-s + (0.343 − 0.220i)6-s + (0.102 − 0.274i)7-s + (−0.0252 + 0.352i)8-s + (0.329 − 0.0474i)9-s + (−0.696 + 0.122i)10-s + (−0.500 − 1.70i)11-s + (−0.138 + 0.253i)12-s + (1.01 − 0.377i)13-s + (0.0295 + 0.205i)14-s + (−0.558 − 0.144i)15-s + (−0.103 − 0.227i)16-s + (−0.843 − 0.183i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.879654 - 0.393033i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.879654 - 0.393033i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.877 - 0.479i)T \) |
| 3 | \( 1 + (0.997 - 0.0713i)T \) |
| 5 | \( 1 + (-2.11 - 0.714i)T \) |
| 23 | \( 1 + (-0.782 - 4.73i)T \) |
| good | 7 | \( 1 + (-0.271 + 0.726i)T + (-5.29 - 4.58i)T^{2} \) |
| 11 | \( 1 + (1.66 + 5.65i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-3.64 + 1.36i)T + (9.82 - 8.51i)T^{2} \) |
| 17 | \( 1 + (3.47 + 0.756i)T + (15.4 + 7.06i)T^{2} \) |
| 19 | \( 1 + (3.10 + 1.99i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (4.42 + 6.87i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (0.455 + 0.525i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-3.93 + 5.26i)T + (-10.4 - 35.5i)T^{2} \) |
| 41 | \( 1 + (-0.0215 + 0.150i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (0.836 + 11.6i)T + (-42.5 + 6.11i)T^{2} \) |
| 47 | \( 1 + (0.875 - 0.875i)T - 47iT^{2} \) |
| 53 | \( 1 + (-8.51 - 3.17i)T + (40.0 + 34.7i)T^{2} \) |
| 59 | \( 1 + (11.6 + 5.33i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-9.88 + 8.56i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-6.73 - 12.3i)T + (-36.2 + 56.3i)T^{2} \) |
| 71 | \( 1 + (-7.40 - 2.17i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (0.107 + 0.492i)T + (-66.4 + 30.3i)T^{2} \) |
| 79 | \( 1 + (-6.48 + 14.1i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-2.80 - 2.09i)T + (23.3 + 79.6i)T^{2} \) |
| 89 | \( 1 + (-0.988 + 1.14i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (3.94 - 2.95i)T + (27.3 - 93.0i)T^{2} \) |
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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
−10.60445829038027106403912402521, −9.409968415976842300178290671235, −8.739503075033546075944015211441, −7.76463051957426010668119677532, −6.66416427409827409225293232885, −5.92311799090596477115798202861, −5.38576769872826260191933174894, −3.75187517915202158080281771411, −2.28969171424853069337852978338, −0.69239391405053173910751472499,
1.52109678001439199215862403586, 2.38477505977994683482121553203, 4.22943557085889015706853487479, 5.12967935580145098092700358501, 6.31304733814454986894673703056, 6.92203728115456688173141952711, 8.199310346171401037864758477560, 9.010848161205769654516557225262, 9.796905632885757035461300264088, 10.53519099224090830091459519873
