L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1.23 − 2.14i)3-s + (−0.499 − 0.866i)4-s + (2.22 + 0.192i)5-s + 2.47·6-s + (−2.62 − 0.300i)7-s + 0.999·8-s + (−1.57 + 2.72i)9-s + (−1.28 + 1.83i)10-s + (−1.96 + 2.66i)11-s + (−1.23 + 2.14i)12-s + 3.09i·13-s + (1.57 − 2.12i)14-s + (−2.34 − 5.02i)15-s + (−0.5 + 0.866i)16-s + (−3.53 + 2.03i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.715 − 1.23i)3-s + (−0.249 − 0.433i)4-s + (0.996 + 0.0859i)5-s + 1.01·6-s + (−0.993 − 0.113i)7-s + 0.353·8-s + (−0.524 + 0.908i)9-s + (−0.404 + 0.579i)10-s + (−0.593 + 0.804i)11-s + (−0.357 + 0.619i)12-s + 0.858i·13-s + (0.420 − 0.568i)14-s + (−0.606 − 1.29i)15-s + (−0.125 + 0.216i)16-s + (−0.856 + 0.494i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0853 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0853 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.379914 + 0.413856i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.379914 + 0.413856i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-2.22 - 0.192i)T \) |
| 7 | \( 1 + (2.62 + 0.300i)T \) |
| 11 | \( 1 + (1.96 - 2.66i)T \) |
good | 3 | \( 1 + (1.23 + 2.14i)T + (-1.5 + 2.59i)T^{2} \) |
| 13 | \( 1 - 3.09iT - 13T^{2} \) |
| 17 | \( 1 + (3.53 - 2.03i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.553 + 0.959i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.755 - 0.435i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.09iT - 29T^{2} \) |
| 31 | \( 1 + (5.80 - 3.35i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.68 - 3.85i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.60T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + (0.969 - 1.67i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (9.95 - 5.74i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.54 - 1.46i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.51 - 2.63i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (12.7 - 7.38i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.20T + 71T^{2} \) |
| 73 | \( 1 + (4.91 - 2.83i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.86 + 1.65i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 8.69iT - 83T^{2} \) |
| 89 | \( 1 + (5.61 + 3.24i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 0.754T + 97T^{2} \) |
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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
−10.48045513564006166451383567348, −9.533182355590052321581504103256, −8.955063768375478979553906187549, −7.53890654080929110764412142388, −6.98167471710918189933689496234, −6.29049440641881325718049590859, −5.72832820857854549152705935777, −4.50281098155723786967089105996, −2.52885147332138221287479709995, −1.40922513802763751894292820863,
0.35283187616878462540466783495, 2.50265516704376668642532619552, 3.45670556551752663559708946642, 4.64794903333496058548582565927, 5.63986454775538983417850167878, 6.16134672911175449058699824478, 7.63056611434652322154796956014, 8.982072876333719004383920687458, 9.436505582918576061042062382194, 10.13799462932428437397354968669