L(s) = 1 | + (0.130 + 0.991i)2-s + (−0.965 + 0.258i)4-s + (0.534 + 1.57i)5-s + (−0.382 − 0.923i)8-s + i·9-s + (−1.49 + 0.735i)10-s + (0.735 − 0.491i)13-s + (0.866 − 0.5i)16-s + (−1.29 − 1.47i)17-s + (−0.991 + 0.130i)18-s + (−0.923 − 1.38i)20-s + (−1.40 + 1.07i)25-s + (0.583 + 0.665i)26-s + (−0.996 + 1.49i)29-s + (0.608 + 0.793i)32-s + ⋯ |
L(s) = 1 | + (0.130 + 0.991i)2-s + (−0.965 + 0.258i)4-s + (0.534 + 1.57i)5-s + (−0.382 − 0.923i)8-s + i·9-s + (−1.49 + 0.735i)10-s + (0.735 − 0.491i)13-s + (0.866 − 0.5i)16-s + (−1.29 − 1.47i)17-s + (−0.991 + 0.130i)18-s + (−0.923 − 1.38i)20-s + (−1.40 + 1.07i)25-s + (0.583 + 0.665i)26-s + (−0.996 + 1.49i)29-s + (0.608 + 0.793i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9791625606\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9791625606\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.130 - 0.991i)T \) |
| 193 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 - iT^{2} \) |
| 5 | \( 1 + (-0.534 - 1.57i)T + (-0.793 + 0.608i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 13 | \( 1 + (-0.735 + 0.491i)T + (0.382 - 0.923i)T^{2} \) |
| 17 | \( 1 + (1.29 + 1.47i)T + (-0.130 + 0.991i)T^{2} \) |
| 19 | \( 1 + (0.793 - 0.608i)T^{2} \) |
| 23 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (0.996 - 1.49i)T + (-0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 + (0.965 - 0.258i)T^{2} \) |
| 37 | \( 1 + (-1.88 + 0.123i)T + (0.991 - 0.130i)T^{2} \) |
| 41 | \( 1 + (-0.0983 + 0.0862i)T + (0.130 - 0.991i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.991 - 0.130i)T^{2} \) |
| 53 | \( 1 + (0.0983 + 1.50i)T + (-0.991 + 0.130i)T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.641 + 1.88i)T + (-0.793 - 0.608i)T^{2} \) |
| 67 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 73 | \( 1 + (-1.31 - 0.0862i)T + (0.991 + 0.130i)T^{2} \) |
| 79 | \( 1 + (-0.793 - 0.608i)T^{2} \) |
| 83 | \( 1 + (0.258 - 0.965i)T^{2} \) |
| 89 | \( 1 + (0.923 + 0.617i)T + (0.382 + 0.923i)T^{2} \) |
| 97 | \( 1 + (0.0675 - 0.513i)T + (-0.965 - 0.258i)T^{2} \) |
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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
−10.93357480717117899671892780437, −9.891224355873371098678576825492, −9.129822099652981403638813921149, −8.019891652489596304233308733427, −7.21905591817230028408200172604, −6.62057390991197871894932295681, −5.71679256196430246747008714727, −4.80320623565642629346082343269, −3.45664185039781494903307179776, −2.40991930193802078733944442266,
1.10002274406697028666163995662, 2.19888845135035083034594986170, 3.99131723526020536489434513401, 4.33645365482320355539558829347, 5.69875976825484371740069226184, 6.25806240902220002330193067062, 8.096387425556686563034044583102, 8.876548989838852650036656968018, 9.257146429404697435112226513179, 10.09511853210015751137610734848