| L(s) = 1 | + (−1.30 − 0.951i)2-s + (0.104 + 0.994i)3-s + (0.190 + 0.587i)4-s + (0.190 + 0.330i)5-s + (0.809 − 1.40i)6-s + (−2.93 − 0.623i)7-s + (−0.690 + 2.12i)8-s + (1.95 − 0.415i)9-s + (0.0646 − 0.614i)10-s + (−3.50 + 3.89i)11-s + (−0.564 + 0.251i)12-s + (−1.69 − 0.754i)13-s + (3.24 + 3.60i)14-s + (−0.309 + 0.224i)15-s + (3.92 − 2.85i)16-s + (−2.83 − 3.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.925 − 0.672i)2-s + (0.0603 + 0.574i)3-s + (0.0954 + 0.293i)4-s + (0.0854 + 0.147i)5-s + (0.330 − 0.572i)6-s + (−1.10 − 0.235i)7-s + (−0.244 + 0.751i)8-s + (0.652 − 0.138i)9-s + (0.0204 − 0.194i)10-s + (−1.05 + 1.17i)11-s + (−0.162 + 0.0725i)12-s + (−0.469 − 0.209i)13-s + (0.868 + 0.964i)14-s + (−0.0797 + 0.0579i)15-s + (0.981 − 0.713i)16-s + (−0.687 − 0.763i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.524406 - 0.409455i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.524406 - 0.409455i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (1.30 + 0.951i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.104 - 0.994i)T + (-2.93 + 0.623i)T^{2} \) |
| 5 | \( 1 + (-0.190 - 0.330i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2.93 + 0.623i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (3.50 - 3.89i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (1.69 + 0.754i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (2.83 + 3.14i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-4.56 + 2.03i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.07 + 3.30i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-5.16 - 3.75i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (-2.11 + 3.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.258 + 2.45i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (2.17 - 0.968i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (-4.54 + 3.30i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (0.692 - 0.147i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.0551 + 0.524i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + (0.118 + 0.204i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.8 + 2.30i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-7.73 + 8.59i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (0.741 - 7.05i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (2.66 + 8.19i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (5.78 + 17.7i)T + (-78.4 + 57.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.801935779498345714592603240408, −9.502813462474698592989271589877, −8.512466627062817682930692567613, −7.32000554161446867742001549941, −6.76515127303165994409979888754, −5.24663295942417957148228231072, −4.56398216439243803534377346877, −3.08305451208115529839210966350, −2.32322323920619696404217915511, −0.53685880572123514149093412181,
0.983392101654024871022079181097, 2.72562334396295455238466597356, 3.76159487124326704021399026823, 5.31562604561284862336206885383, 6.30415984524122667901508909135, 6.90069212823757409741792162584, 7.82323145720197263930640418511, 8.317388484345239503312277734444, 9.350929395198442328631994373682, 9.859608440764023695335540302838
