L(s) = 1 | − i·2-s + (−1.37 + 1.05i)3-s − 4-s − 2.38·5-s + (1.05 + 1.37i)6-s + (−2.59 − 0.527i)7-s + i·8-s + (0.764 − 2.90i)9-s + 2.38i·10-s − 4.24i·11-s + (1.37 − 1.05i)12-s − 0.530i·13-s + (−0.527 + 2.59i)14-s + (3.27 − 2.52i)15-s + 16-s − 2.11·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.792 + 0.610i)3-s − 0.5·4-s − 1.06·5-s + (0.431 + 0.560i)6-s + (−0.979 − 0.199i)7-s + 0.353i·8-s + (0.254 − 0.966i)9-s + 0.754i·10-s − 1.28i·11-s + (0.396 − 0.305i)12-s − 0.147i·13-s + (−0.141 + 0.692i)14-s + (0.845 − 0.651i)15-s + 0.250·16-s − 0.512·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.461352 + 0.106954i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.461352 + 0.106954i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (1.37 - 1.05i)T \) |
| 7 | \( 1 + (2.59 + 0.527i)T \) |
| 23 | \( 1 + iT \) |
good | 5 | \( 1 + 2.38T + 5T^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 + 0.530iT - 13T^{2} \) |
| 17 | \( 1 + 2.11T + 17T^{2} \) |
| 19 | \( 1 - 2.73iT - 19T^{2} \) |
| 29 | \( 1 - 4.13iT - 29T^{2} \) |
| 31 | \( 1 - 8.63iT - 31T^{2} \) |
| 37 | \( 1 - 1.99T + 37T^{2} \) |
| 41 | \( 1 - 8.92T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 + 5.75T + 47T^{2} \) |
| 53 | \( 1 - 2.38iT - 53T^{2} \) |
| 59 | \( 1 + 4.97T + 59T^{2} \) |
| 61 | \( 1 + 3.14iT - 61T^{2} \) |
| 67 | \( 1 + 7.15T + 67T^{2} \) |
| 71 | \( 1 - 11.1iT - 71T^{2} \) |
| 73 | \( 1 + 5.68iT - 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 1.92T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 + 9.07iT - 97T^{2} \) |
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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
−10.40774463075123231188015489022, −9.341159159946542834271287165551, −8.703930061868199661223330579876, −7.61335584782663401457786230149, −6.49065319238461178267851007469, −5.70475826797281226527287944571, −4.54081377632512849495808275065, −3.69102865195109838068734448536, −3.09137157206852398246402820413, −0.797866506778339638009561802683,
0.37344150622309600804601073170, 2.38298362672689134189025876352, 4.02762994535554171535195356611, 4.69300572507160577799477121926, 5.92058790488666729063673792046, 6.57698916233695075966472274878, 7.47867538264448858978166556296, 7.78330862790430065862663407216, 9.130109295117636340755132350075, 9.806390363146128908531804473950