| L(s) = 1 | + (−2.82 + 2.82i)2-s − 16.0i·4-s + (−117. − 117. i)7-s + (45.2 + 45.2i)8-s + 28.2i·11-s + (−283. + 283. i)13-s + 662.·14-s − 256.·16-s + (137. − 137. i)17-s + 2.80e3i·19-s + (−79.9 − 79.9i)22-s + (902. + 902. i)23-s − 1.60e3i·26-s + (−1.87e3 + 1.87e3i)28-s − 827.·29-s + ⋯ |
| L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.903 − 0.903i)7-s + (0.250 + 0.250i)8-s + 0.0704i·11-s + (−0.464 + 0.464i)13-s + 0.903·14-s − 0.250·16-s + (0.115 − 0.115i)17-s + 1.78i·19-s + (−0.0352 − 0.0352i)22-s + (0.355 + 0.355i)23-s − 0.464i·26-s + (−0.451 + 0.451i)28-s − 0.182·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8984867111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8984867111\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.82 - 2.82i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (117. + 117. i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 28.2iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (283. - 283. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (-137. + 137. i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 2.80e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-902. - 902. i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 827.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-1.32e3 - 1.32e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.07e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-7.83e3 + 7.83e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (-1.17e4 + 1.17e4i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (2.25e4 + 2.25e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + 3.31e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (2.74e4 + 2.74e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 5.46e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (1.42e4 - 1.42e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 7.92e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (3.65e4 + 3.65e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 6.99e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (6.93e4 + 6.93e4i)T + 8.58e9iT^{2} \) |
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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
−9.965725426299719181684658705222, −9.463735908400984543319530635858, −8.241327048591174266130456766900, −7.38939953331552228607991440342, −6.62044000668219433054680627373, −5.68565190505390034741219687979, −4.37004969140255570452392371331, −3.29272345507054464777589884990, −1.66532719152106014221194358014, −0.35703893432034945140823335646,
0.77128941195205316534657454850, 2.45126095477426295105983209981, 3.05202215415637436044275561789, 4.52767444987573213071152726127, 5.73610955911808004170164908865, 6.77314919800301221701219766590, 7.75898824338598750471449775614, 8.986139824560553415941494326596, 9.302942338229337174104133009243, 10.38147805276089491487567695643
