| L(s) = 1 | + (−0.483 + 1.32i)2-s + (−0.00732 − 0.0143i)3-s + (−1.53 − 1.28i)4-s + (−2.23 + 0.100i)5-s + (0.0226 − 0.00278i)6-s + (−0.707 − 0.707i)7-s + (2.44 − 1.41i)8-s + (1.76 − 2.42i)9-s + (0.947 − 3.01i)10-s + (1.01 + 1.39i)11-s + (−0.00726 + 0.0314i)12-s + (0.349 + 2.20i)13-s + (1.28 − 0.597i)14-s + (0.0178 + 0.0313i)15-s + (0.695 + 3.93i)16-s + (−2.42 − 1.23i)17-s + ⋯ |
| L(s) = 1 | + (−0.341 + 0.939i)2-s + (−0.00423 − 0.00830i)3-s + (−0.766 − 0.642i)4-s + (−0.998 + 0.0448i)5-s + (0.00925 − 0.00113i)6-s + (−0.267 − 0.267i)7-s + (0.865 − 0.500i)8-s + (0.587 − 0.808i)9-s + (0.299 − 0.954i)10-s + (0.306 + 0.421i)11-s + (−0.00209 + 0.00908i)12-s + (0.0969 + 0.612i)13-s + (0.342 − 0.159i)14-s + (0.00459 + 0.00810i)15-s + (0.173 + 0.984i)16-s + (−0.588 − 0.299i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.355965 + 0.688894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.355965 + 0.688894i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.483 - 1.32i)T \) |
| 5 | \( 1 + (2.23 - 0.100i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| good | 3 | \( 1 + (0.00732 + 0.0143i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (-1.01 - 1.39i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.349 - 2.20i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (2.42 + 1.23i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.57 - 7.91i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.352 - 2.22i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-3.47 - 1.12i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (9.19 - 2.98i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (3.89 - 0.617i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-6.44 - 4.68i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (3.18 - 3.18i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.782 + 0.398i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-7.79 + 3.97i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-10.9 - 7.93i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (1.02 - 0.745i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (2.89 - 5.67i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-12.6 - 4.12i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (10.0 + 1.58i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (3.89 - 11.9i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-10.6 - 5.44i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (1.96 + 2.70i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.73 - 11.2i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.49583219091906508317019455303, −9.693583044343422198477406812249, −8.965550641256955951179446494541, −8.024285969326355677929464268718, −7.12095276738078948012077184603, −6.71474181095541768597548344886, −5.48739856136971045664059103136, −4.21867889753322268244964859459, −3.68792009019783539553840144931, −1.28298567731308014204602406263,
0.53160410981258939874449970257, 2.29770643293230022165258824226, 3.42183300915223089007021034395, 4.37766497823196733687959846315, 5.28398844667290852973610409869, 6.95552205341209014909255075459, 7.69623646265028535449005036701, 8.640506535398879043860577958169, 9.208190402666889831114628441759, 10.41645079268964062409915369740
