L(s) = 1 | + (0.608 + 0.793i)2-s + (−0.258 + 0.965i)4-s + (−1.57 − 0.534i)5-s + (−0.923 + 0.382i)8-s + (0.991 + 0.130i)9-s + (−0.534 − 1.57i)10-s + (1 + i)13-s + (−0.866 − 0.499i)16-s + (−0.991 + 0.130i)17-s + (0.499 + 0.866i)18-s + (0.923 − 1.38i)20-s + (1.40 + 1.07i)25-s + (−0.184 + 1.40i)26-s + (0.216 + 1.08i)29-s + (−0.130 − 0.991i)32-s + ⋯ |
L(s) = 1 | + (0.608 + 0.793i)2-s + (−0.258 + 0.965i)4-s + (−1.57 − 0.534i)5-s + (−0.923 + 0.382i)8-s + (0.991 + 0.130i)9-s + (−0.534 − 1.57i)10-s + (1 + i)13-s + (−0.866 − 0.499i)16-s + (−0.991 + 0.130i)17-s + (0.499 + 0.866i)18-s + (0.923 − 1.38i)20-s + (1.40 + 1.07i)25-s + (−0.184 + 1.40i)26-s + (0.216 + 1.08i)29-s + (−0.130 − 0.991i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.019953142\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.019953142\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.608 - 0.793i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (0.991 - 0.130i)T \) |
good | 3 | \( 1 + (-0.991 - 0.130i)T^{2} \) |
| 5 | \( 1 + (1.57 + 0.534i)T + (0.793 + 0.608i)T^{2} \) |
| 11 | \( 1 + (0.608 + 0.793i)T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 19 | \( 1 + (-0.258 + 0.965i)T^{2} \) |
| 23 | \( 1 + (0.991 - 0.130i)T^{2} \) |
| 29 | \( 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + (0.991 + 0.130i)T^{2} \) |
| 37 | \( 1 + (1.49 - 0.735i)T + (0.608 - 0.793i)T^{2} \) |
| 41 | \( 1 + (0.382 - 1.92i)T + (-0.923 - 0.382i)T^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (1.83 - 0.241i)T + (0.965 - 0.258i)T^{2} \) |
| 59 | \( 1 + (0.258 + 0.965i)T^{2} \) |
| 61 | \( 1 + (0.257 - 0.293i)T + (-0.130 - 0.991i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 73 | \( 1 + (-0.835 + 0.732i)T + (0.130 - 0.991i)T^{2} \) |
| 79 | \( 1 + (-0.991 + 0.130i)T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + (-0.517 - 1.93i)T + (-0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (-1.08 + 0.216i)T + (0.923 - 0.382i)T^{2} \) |
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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
−8.775050175558192091102999055482, −8.262288923278306901683166613874, −7.60365134332408512597958368133, −6.78027123597404370762440183284, −6.44054502107909893516849960359, −4.98152608934202298939078524194, −4.57065101873150678243219154665, −3.89867032184114534238265491262, −3.23000902679964912236003364736, −1.53618531007997584947266727255,
0.51903379653606829551190515665, 1.95582117272900423141083041928, 3.16894510476533483541059624246, 3.76193222846466822117350153403, 4.30674238774259270617469737270, 5.19612697603967717619097771212, 6.27865728976188979159842299805, 6.94444875503331023608110847973, 7.72248363385291298282341487677, 8.515870539047614083985435393724