L(s) = 1 | + 2.23i·2-s − 3.00·4-s + (2.23 − 2.23i)7-s − 2.23i·8-s − 3·9-s + (4 − 4i)11-s + (4.47 − 4.47i)13-s + (5.00 + 5.00i)14-s − 0.999·16-s − 4.47i·17-s − 6.70i·18-s + (−2 − 2i)19-s + (8.94 + 8.94i)22-s + (−2.23 − 2.23i)23-s + (10.0 + 10.0i)26-s + ⋯ |
L(s) = 1 | + 1.58i·2-s − 1.50·4-s + (0.845 − 0.845i)7-s − 0.790i·8-s − 9-s + (1.20 − 1.20i)11-s + (1.24 − 1.24i)13-s + (1.33 + 1.33i)14-s − 0.249·16-s − 1.08i·17-s − 1.58i·18-s + (−0.458 − 0.458i)19-s + (1.90 + 1.90i)22-s + (−0.466 − 0.466i)23-s + (1.96 + 1.96i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37732 + 0.537249i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37732 + 0.537249i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 + (-2 - 5i)T \) |
good | 2 | \( 1 - 2.23iT - 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 + (-2.23 + 2.23i)T - 7iT^{2} \) |
| 11 | \( 1 + (-4 + 4i)T - 11iT^{2} \) |
| 13 | \( 1 + (-4.47 + 4.47i)T - 13iT^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 + (2 + 2i)T + 19iT^{2} \) |
| 23 | \( 1 + (2.23 + 2.23i)T + 23iT^{2} \) |
| 31 | \( 1 + (4 - 4i)T - 31iT^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + (-1 - i)T + 41iT^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 - 4.47T + 47T^{2} \) |
| 53 | \( 1 + (-8.94 - 8.94i)T + 53iT^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + (9 - 9i)T - 61iT^{2} \) |
| 67 | \( 1 + (2.23 + 2.23i)T + 67iT^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 - 4.47iT - 73T^{2} \) |
| 79 | \( 1 + (2 + 2i)T + 79iT^{2} \) |
| 83 | \( 1 + (-6.70 - 6.70i)T + 83iT^{2} \) |
| 89 | \( 1 + (7 + 7i)T + 89iT^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−10.74136134416603112075518368387, −8.887086884155050082388822178901, −8.752768244894637667899420099515, −7.85717625760770281295455458939, −6.99458235093351901978975708257, −6.05898299075652317881794078601, −5.47590273613522818991366041855, −4.35750939959986218037553529475, −3.20244937692115947667111500139, −0.837137830147755927829404682578,
1.65878360496941161617950942535, 2.13988931774515426549444102533, 3.78219528517658491590116549786, 4.27906155687111406037939896444, 5.68462271901943191206588834853, 6.61777229723058957487742732744, 8.191643950684312315674791011503, 8.912967775440025751294601854481, 9.444387226104391380215873622585, 10.54579075916469329177728812553