L(s) = 1 | + (−0.991 + 0.130i)2-s + (0.965 − 0.258i)4-s + (−1.25 + 1.09i)5-s + (−0.923 + 0.382i)8-s + (−0.608 + 0.793i)9-s + (1.09 − 1.25i)10-s + (−1 − i)13-s + (0.866 − 0.5i)16-s + (−0.608 − 0.793i)17-s + (0.499 − 0.866i)18-s + (−0.923 + 1.38i)20-s + (0.230 − 1.75i)25-s + (1.12 + 0.860i)26-s + (0.216 + 1.08i)29-s + (−0.793 + 0.608i)32-s + ⋯ |
L(s) = 1 | + (−0.991 + 0.130i)2-s + (0.965 − 0.258i)4-s + (−1.25 + 1.09i)5-s + (−0.923 + 0.382i)8-s + (−0.608 + 0.793i)9-s + (1.09 − 1.25i)10-s + (−1 − i)13-s + (0.866 − 0.5i)16-s + (−0.608 − 0.793i)17-s + (0.499 − 0.866i)18-s + (−0.923 + 1.38i)20-s + (0.230 − 1.75i)25-s + (1.12 + 0.860i)26-s + (0.216 + 1.08i)29-s + (−0.793 + 0.608i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3302086569\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3302086569\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.991 - 0.130i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (0.608 + 0.793i)T \) |
good | 3 | \( 1 + (0.608 - 0.793i)T^{2} \) |
| 5 | \( 1 + (1.25 - 1.09i)T + (0.130 - 0.991i)T^{2} \) |
| 11 | \( 1 + (-0.991 + 0.130i)T^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 19 | \( 1 + (0.965 - 0.258i)T^{2} \) |
| 23 | \( 1 + (-0.608 - 0.793i)T^{2} \) |
| 29 | \( 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + (-0.608 + 0.793i)T^{2} \) |
| 37 | \( 1 + (-0.108 + 1.65i)T + (-0.991 - 0.130i)T^{2} \) |
| 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 | \( 1 + (-1.12 - 1.46i)T + (-0.258 + 0.965i)T^{2} \) |
| 59 | \( 1 + (-0.965 - 0.258i)T^{2} \) |
| 61 | \( 1 + (-0.125 - 0.369i)T + (-0.793 + 0.608i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 73 | \( 1 + (-1.05 - 0.357i)T + (0.793 + 0.608i)T^{2} \) |
| 79 | \( 1 + (0.608 + 0.793i)T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + (-1.93 - 0.517i)T + (0.866 + 0.5i)T^{2} \) |
| 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
−8.604872396194245676337055368207, −7.85317246391225362342762779653, −7.29221372835789240876587583743, −7.01098582950715091906663172745, −5.80618734945098863253887551084, −5.06808029263947833289048531946, −3.81063381224510067285138723935, −2.80160115307644440072225829674, −2.37714822599682249856687632883, −0.34304313827041487629620146308,
0.923812881291942122434382030026, 2.19001516169376168124004555707, 3.34701230726544895175178368561, 4.18793765781909590994019372659, 4.93780081097358880101804459383, 6.21095265060033767716301640564, 6.76720550184861625127998067036, 7.74245973721870899435622763040, 8.263820670059894939065407642842, 8.790891383705857686759402000649