L(s) = 1 | − 2.73·3-s − 0.732i·5-s + (2 − 1.73i)7-s + 4.46·9-s − 5.46i·11-s − 4.73i·13-s + 2i·15-s + 4i·17-s − 1.26·19-s + (−5.46 + 4.73i)21-s − 5.46i·23-s + 4.46·25-s − 3.99·27-s + 6.92·29-s − 6.92·31-s + ⋯ |
L(s) = 1 | − 1.57·3-s − 0.327i·5-s + (0.755 − 0.654i)7-s + 1.48·9-s − 1.64i·11-s − 1.31i·13-s + 0.516i·15-s + 0.970i·17-s − 0.290·19-s + (−1.19 + 1.03i)21-s − 1.13i·23-s + 0.892·25-s − 0.769·27-s + 1.28·29-s − 1.24·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8492013368\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8492013368\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 + 0.732iT - 5T^{2} \) |
| 11 | \( 1 + 5.46iT - 11T^{2} \) |
| 13 | \( 1 + 4.73iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 + 5.46iT - 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 2.92iT - 41T^{2} \) |
| 43 | \( 1 - 2.53iT - 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 6.92T + 53T^{2} \) |
| 59 | \( 1 + 3.80T + 59T^{2} \) |
| 61 | \( 1 - 11.6iT - 61T^{2} \) |
| 67 | \( 1 - 2.53iT - 67T^{2} \) |
| 71 | \( 1 + 4.53iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 14.9iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−8.733670139958558873845809613913, −8.257433710272287335165242865605, −7.32029652069656421234304766321, −6.22010987074733503144107211328, −5.85963349424013537739686565836, −4.96698326196732127796127821997, −4.28588650860269362416026941410, −3.05879927873469591840254658431, −1.20892283017397454058010873368, −0.46713505501031815639033187369,
1.42321608786732797968578676570, 2.41170295524548352587467174970, 4.16160377185298381923083526477, 4.89376840711853938169698987605, 5.36630577730937644251720195305, 6.49439744734976087402247218529, 6.97769629425799622318549674531, 7.71898137900520945022078735164, 9.067204340804954309481435986244, 9.593827513361540581722165643268