L(s) = 1 | − 0.517·3-s + (3.34 + 3.34i)7-s − 2.73·9-s + (1.09 − 1.09i)11-s − 4.89i·13-s + (−0.707 − 0.707i)17-s + (2.09 − 2.09i)19-s + (−1.73 − 1.73i)21-s + (4.38 − 4.38i)23-s + 2.96·27-s + (4.73 + 4.73i)29-s + 6.19i·31-s + (−0.568 + 0.568i)33-s − 6.03i·37-s + 2.53i·39-s + ⋯ |
L(s) = 1 | − 0.298·3-s + (1.26 + 1.26i)7-s − 0.910·9-s + (0.331 − 0.331i)11-s − 1.35i·13-s + (−0.171 − 0.171i)17-s + (0.481 − 0.481i)19-s + (−0.377 − 0.377i)21-s + (0.913 − 0.913i)23-s + 0.571·27-s + (0.878 + 0.878i)29-s + 1.11i·31-s + (−0.0989 + 0.0989i)33-s − 0.992i·37-s + 0.406i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.720050356\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.720050356\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 0.517T + 3T^{2} \) |
| 7 | \( 1 + (-3.34 - 3.34i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.09 + 1.09i)T - 11iT^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + (0.707 + 0.707i)T + 17iT^{2} \) |
| 19 | \( 1 + (-2.09 + 2.09i)T - 19iT^{2} \) |
| 23 | \( 1 + (-4.38 + 4.38i)T - 23iT^{2} \) |
| 29 | \( 1 + (-4.73 - 4.73i)T + 29iT^{2} \) |
| 31 | \( 1 - 6.19iT - 31T^{2} \) |
| 37 | \( 1 + 6.03iT - 37T^{2} \) |
| 41 | \( 1 + 0.464iT - 41T^{2} \) |
| 43 | \( 1 + 0.656iT - 43T^{2} \) |
| 47 | \( 1 + (1.41 - 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 - 9.89T + 53T^{2} \) |
| 59 | \( 1 + (-7.73 - 7.73i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.19 - 3.19i)T - 61iT^{2} \) |
| 67 | \( 1 - 5.79iT - 67T^{2} \) |
| 71 | \( 1 + 0.928T + 71T^{2} \) |
| 73 | \( 1 + (8.81 + 8.81i)T + 73iT^{2} \) |
| 79 | \( 1 + 2.19T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + (-11.5 - 11.5i)T + 97iT^{2} \) |
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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
−8.966932756592556448177318661101, −8.771451416839679569645453455053, −8.029727786199908051425537598015, −7.00589223124035210004189321889, −5.90672578935001947897358359997, −5.32552416253993979047533670211, −4.76452960338871119873881043097, −3.15056928884928739835857852930, −2.46550597921708030836803799769, −0.943544569882306147760475535496,
1.01158589863644081046953561249, 2.09448334938125941112655957791, 3.61280885615848798897063556415, 4.45607783224805646418774903214, 5.11832908308066297510711140148, 6.23656250507208297422897660678, 7.03309192636675963712768629241, 7.79626291449334298686531402931, 8.509506242186773457736026409690, 9.431203312849332684937343685874