L(s) = 1 | + (0.866 − 0.5i)3-s + (−1.61 + 2.79i)5-s + (1.82 + 1.91i)7-s + (0.499 − 0.866i)9-s + (1.10 + 1.91i)11-s − 5.08·13-s + 3.22i·15-s + (−2.73 + 1.57i)17-s + (−2.93 − 1.69i)19-s + (2.53 + 0.743i)21-s + (2.65 + 1.53i)23-s + (−2.70 − 4.69i)25-s − 0.999i·27-s + 9.88i·29-s + (−1.01 − 1.75i)31-s + ⋯ |
L(s) = 1 | + (0.499 − 0.288i)3-s + (−0.721 + 1.25i)5-s + (0.690 + 0.723i)7-s + (0.166 − 0.288i)9-s + (0.333 + 0.577i)11-s − 1.40·13-s + 0.833i·15-s + (−0.663 + 0.383i)17-s + (−0.673 − 0.388i)19-s + (0.554 + 0.162i)21-s + (0.553 + 0.319i)23-s + (−0.541 − 0.938i)25-s − 0.192i·27-s + 1.83i·29-s + (−0.182 − 0.315i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.872656 + 1.02488i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.872656 + 1.02488i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-1.82 - 1.91i)T \) |
good | 5 | \( 1 + (1.61 - 2.79i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.10 - 1.91i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.08T + 13T^{2} \) |
| 17 | \( 1 + (2.73 - 1.57i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.93 + 1.69i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.65 - 1.53i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 9.88iT - 29T^{2} \) |
| 31 | \( 1 + (1.01 + 1.75i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.798 - 0.460i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 5.96iT - 41T^{2} \) |
| 43 | \( 1 - 6.68T + 43T^{2} \) |
| 47 | \( 1 + (-1.06 + 1.83i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.12 + 1.80i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-10.6 + 6.14i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.34 + 10.9i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.40 - 7.63i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 11.7iT - 71T^{2} \) |
| 73 | \( 1 + (-7.82 + 4.51i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (9.60 + 5.54i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.57iT - 83T^{2} \) |
| 89 | \( 1 + (6.32 + 3.65i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 15.2iT - 97T^{2} \) |
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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
−10.91595663483597840868931195662, −9.856070339120374869786799475695, −8.925641883543172800990404674976, −8.055592090037272855102281372910, −7.14190334776766716765099117555, −6.73108334700652250745133526993, −5.21075913774676806229180203957, −4.14842420470758862207582084321, −2.89866604823846143094518822478, −2.06798336969413780908391551939,
0.66837890874171920536640087920, 2.34624340714043343681091241146, 4.04550615406268352824166319079, 4.45326996717454895113871991865, 5.43768099531230292448406230137, 7.01310079169772257469150546898, 7.84170204540177237117035318701, 8.516772130040945026707905874954, 9.234161149595472915275488725507, 10.20094012845577752519118491649