L(s) = 1 | + (0.666 − 1.59i)3-s − 0.468·5-s + (−2.39 + 1.13i)7-s + (−2.11 − 2.12i)9-s − 1.34·11-s + (−3.16 − 5.48i)13-s + (−0.311 + 0.748i)15-s + (−2.47 − 4.28i)17-s + (2.38 − 4.13i)19-s + (0.222 + 4.57i)21-s + 7.62·23-s − 4.78·25-s + (−4.81 + 1.95i)27-s + (−1.80 + 3.12i)29-s + (−3.24 + 5.62i)31-s + ⋯ |
L(s) = 1 | + (0.384 − 0.923i)3-s − 0.209·5-s + (−0.903 + 0.428i)7-s + (−0.704 − 0.709i)9-s − 0.406·11-s + (−0.877 − 1.52i)13-s + (−0.0805 + 0.193i)15-s + (−0.599 − 1.03i)17-s + (0.548 − 0.949i)19-s + (0.0485 + 0.998i)21-s + 1.59·23-s − 0.956·25-s + (−0.926 + 0.377i)27-s + (−0.335 + 0.580i)29-s + (−0.583 + 1.01i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.831 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.259614 - 0.854598i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.259614 - 0.854598i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.666 + 1.59i)T \) |
| 7 | \( 1 + (2.39 - 1.13i)T \) |
good | 5 | \( 1 + 0.468T + 5T^{2} \) |
| 11 | \( 1 + 1.34T + 11T^{2} \) |
| 13 | \( 1 + (3.16 + 5.48i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.47 + 4.28i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.38 + 4.13i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 7.62T + 23T^{2} \) |
| 29 | \( 1 + (1.80 - 3.12i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.24 - 5.62i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.24 + 9.07i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0251 + 0.0435i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.431 - 0.748i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.49 - 9.51i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.84 - 10.1i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.93 + 3.35i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.87 + 3.24i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.32 + 2.29i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.04T + 71T^{2} \) |
| 73 | \( 1 + (3.30 + 5.71i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.58 + 2.75i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.90 + 8.49i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.30 + 9.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.97 + 12.0i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.62393157371201042453126607987, −9.354078911005655219193108600670, −8.928022296430133581397767203012, −7.45140332146868664501302679990, −7.29711869842743248569836578432, −5.94992962873295398596727283690, −5.04603922300533493962171097879, −3.15914691525975431270270651564, −2.60886875618298855547463599865, −0.47973028307300541520078393091,
2.31550880145490116684722383587, 3.66364804165151032914956567720, 4.34574272046265913481620108220, 5.58338806589948007641365977963, 6.76038817545203448192225211410, 7.73122326844200839757645731216, 8.804327551878322633338225966542, 9.651566037486263455706650239049, 10.14206535067015832386450725821, 11.19563634384176419637905735910