| L(s) = 1 | + (−0.482 + 1.32i)2-s + (0.891 − 0.453i)3-s + (−1.53 − 1.28i)4-s + (2.23 − 0.0415i)5-s + (0.174 + 1.40i)6-s + (−1.15 + 1.15i)7-s + (2.44 − 1.42i)8-s + (0.587 − 0.809i)9-s + (−1.02 + 2.99i)10-s + (0.574 + 0.790i)11-s + (−1.94 − 0.445i)12-s + (3.66 − 0.580i)13-s + (−0.980 − 2.09i)14-s + (1.97 − 1.05i)15-s + (0.714 + 3.93i)16-s + (1.54 − 3.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.340 + 0.940i)2-s + (0.514 − 0.262i)3-s + (−0.767 − 0.640i)4-s + (0.999 − 0.0186i)5-s + (0.0710 + 0.572i)6-s + (−0.437 + 0.437i)7-s + (0.864 − 0.503i)8-s + (0.195 − 0.269i)9-s + (−0.323 + 0.946i)10-s + (0.173 + 0.238i)11-s + (−0.562 − 0.128i)12-s + (1.01 − 0.161i)13-s + (−0.261 − 0.559i)14-s + (0.509 − 0.271i)15-s + (0.178 + 0.983i)16-s + (0.374 − 0.734i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27398 + 0.600497i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27398 + 0.600497i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.482 - 1.32i)T \) |
| 3 | \( 1 + (-0.891 + 0.453i)T \) |
| 5 | \( 1 + (-2.23 + 0.0415i)T \) |
| good | 7 | \( 1 + (1.15 - 1.15i)T - 7iT^{2} \) |
| 11 | \( 1 + (-0.574 - 0.790i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.66 + 0.580i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.54 + 3.02i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.54 - 4.75i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.65 - 0.262i)T + (21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (6.16 + 2.00i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (3.69 - 1.20i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.23 - 7.77i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (2.58 + 1.87i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (6.18 + 6.18i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.67 + 9.17i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (0.989 + 1.94i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-2.96 - 2.15i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (3.48 - 2.52i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (6.00 + 3.05i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-2.89 - 0.939i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.20 + 13.9i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (0.430 - 1.32i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-7.43 + 14.5i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-1.66 - 2.29i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (12.3 - 6.30i)T + (57.0 - 78.4i)T^{2} \) |
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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
−12.06069951451807837232309664470, −10.52357081703391397972866050859, −9.649062864874449124599397163794, −9.039119133229105619868406809691, −8.090770242241168791481197213004, −6.96145393131808276363547847115, −6.06892885827006534217911486033, −5.22373442947527525980150615606, −3.47295845130213854229580134417, −1.62069956474741935359451097423,
1.50590242904446924058883002714, 2.97766834662985116215596536213, 3.98839883300187515233178151730, 5.42805472029717431363106292233, 6.81346066294914463464101330629, 8.145565497245365023832221477751, 9.171434821347740894894899190631, 9.615808554060767703721962463749, 10.70411552252592067130546315304, 11.23143820206250999119654875594
