| L(s) = 1 | + (−1.41 + 0.0483i)2-s + (−0.993 − 1.94i)3-s + (1.99 − 0.136i)4-s + (−0.0255 + 2.23i)5-s + (1.49 + 2.70i)6-s + (0.707 + 0.707i)7-s + (−2.81 + 0.289i)8-s + (−1.04 + 1.44i)9-s + (−0.0719 − 3.16i)10-s + (−0.720 − 0.991i)11-s + (−2.24 − 3.75i)12-s + (−0.598 − 3.77i)13-s + (−1.03 − 0.965i)14-s + (4.38 − 2.17i)15-s + (3.96 − 0.545i)16-s + (0.871 + 0.444i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 + 0.0341i)2-s + (−0.573 − 1.12i)3-s + (0.997 − 0.0683i)4-s + (−0.0114 + 0.999i)5-s + (0.611 + 1.10i)6-s + (0.267 + 0.267i)7-s + (−0.994 + 0.102i)8-s + (−0.349 + 0.481i)9-s + (−0.0227 − 0.999i)10-s + (−0.217 − 0.298i)11-s + (−0.648 − 1.08i)12-s + (−0.165 − 1.04i)13-s + (−0.276 − 0.257i)14-s + (1.13 − 0.560i)15-s + (0.990 − 0.136i)16-s + (0.211 + 0.107i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.786095 - 0.118823i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.786095 - 0.118823i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.0483i)T \) |
| 5 | \( 1 + (0.0255 - 2.23i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| good | 3 | \( 1 + (0.993 + 1.94i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (0.720 + 0.991i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.598 + 3.77i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.871 - 0.444i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.07 - 6.37i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.887 - 5.60i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-8.54 - 2.77i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.40 + 2.08i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-7.20 + 1.14i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (2.05 + 1.49i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-6.58 + 6.58i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.31 + 0.667i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-2.94 + 1.50i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-5.67 - 4.12i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.07 + 1.50i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-5.09 + 9.99i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (3.11 + 1.01i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (3.90 + 0.619i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (2.41 - 7.42i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.861 + 0.439i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (4.36 + 6.00i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.07 - 9.95i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.34509493633177645502698391962, −9.799667388366432032824307505541, −8.329522089815601379788011342357, −7.74248344041173349357407265810, −7.11208231584996717528912206313, −6.07604593135033332540194676277, −5.66092540962432517348592915809, −3.38278575332639058219964934130, −2.28091973544424253278781852927, −0.968271487825205135972905805897,
0.860918251304238539272036991343, 2.56378057133169710215373747045, 4.38150246895367415168330947646, 4.78276243781426730676131629729, 6.05539725033343159099670155445, 7.08541898171418799832646853243, 8.190668027600687936443613716922, 8.933685529760620147225021875229, 9.761495826701159050159854873890, 10.19442281359922345593300981010
