| L(s) = 1 | + (−0.455 + 1.33i)2-s + (−0.432 − 0.197i)3-s + (−1.58 − 1.21i)4-s + (−0.962 + 3.27i)5-s + (0.461 − 0.489i)6-s + (1.17 − 3.38i)7-s + (2.35 − 1.56i)8-s + (−1.81 − 2.09i)9-s + (−3.95 − 2.78i)10-s + (3.29 − 0.800i)11-s + (0.445 + 0.841i)12-s + (−1.16 − 2.25i)13-s + (3.99 + 3.10i)14-s + (1.06 − 1.22i)15-s + (1.03 + 3.86i)16-s + (3.73 + 1.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 + 0.946i)2-s + (−0.249 − 0.114i)3-s + (−0.792 − 0.609i)4-s + (−0.430 + 1.46i)5-s + (0.188 − 0.199i)6-s + (0.442 − 1.27i)7-s + (0.831 − 0.554i)8-s + (−0.605 − 0.698i)9-s + (−1.24 − 0.879i)10-s + (0.994 − 0.241i)11-s + (0.128 + 0.242i)12-s + (−0.323 − 0.626i)13-s + (1.06 + 0.830i)14-s + (0.275 − 0.317i)15-s + (0.257 + 0.966i)16-s + (0.905 + 0.362i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.957462 + 0.223597i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.957462 + 0.223597i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.455 - 1.33i)T \) |
| 67 | \( 1 + (4.81 + 6.62i)T \) |
| good | 3 | \( 1 + (0.432 + 0.197i)T + (1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (0.962 - 3.27i)T + (-4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-1.17 + 3.38i)T + (-5.50 - 4.32i)T^{2} \) |
| 11 | \( 1 + (-3.29 + 0.800i)T + (9.77 - 5.04i)T^{2} \) |
| 13 | \( 1 + (1.16 + 2.25i)T + (-7.54 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-3.73 - 1.49i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-2.38 + 0.826i)T + (14.9 - 11.7i)T^{2} \) |
| 23 | \( 1 + (-0.379 - 0.533i)T + (-7.52 + 21.7i)T^{2} \) |
| 29 | \( 1 + (-0.484 + 0.279i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.88 + 1.48i)T + (17.9 + 25.2i)T^{2} \) |
| 37 | \( 1 + (-4.93 - 2.84i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.07 + 5.56i)T + (9.66 - 39.8i)T^{2} \) |
| 43 | \( 1 + (-8.33 + 1.19i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (-5.87 + 0.560i)T + (46.1 - 8.89i)T^{2} \) |
| 53 | \( 1 + (9.07 + 1.30i)T + (50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-3.26 - 5.08i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-0.499 - 0.121i)T + (54.2 + 27.9i)T^{2} \) |
| 71 | \( 1 + (-6.82 + 2.73i)T + (51.3 - 48.9i)T^{2} \) |
| 73 | \( 1 + (-2.54 + 10.5i)T + (-64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (0.546 - 11.4i)T + (-78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (4.77 + 5.00i)T + (-3.94 + 82.9i)T^{2} \) |
| 89 | \( 1 + (7.38 + 16.1i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-7.67 + 13.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.81839045534488018054160258140, −10.05140566548002313724722247342, −9.073087537298070436699750739636, −7.73388072910468579988904484369, −7.38573201540933856822268300160, −6.47990099402533124274396833137, −5.70811886249314137608757436064, −4.16261677005023958661747780689, −3.31640293512347058706793954439, −0.833943331213359273679167896688,
1.22577199050696799375051656817, 2.52209040949857767342290068264, 4.12591264637989480946619441675, 4.97280957039278979549437729350, 5.68694458579338087012135941119, 7.64671572146131196676590184129, 8.382043395740172155393260637957, 9.172596653916580684106133544139, 9.567898733768653698185402466280, 11.10393392877362039911348466897
