| L(s) = 1 | + (0.951 − 0.309i)2-s + (−0.974 − 1.34i)3-s + (0.809 − 0.587i)4-s + (2.09 − 0.783i)5-s + (−1.34 − 0.974i)6-s + i·7-s + (0.587 − 0.809i)8-s + (0.0768 − 0.236i)9-s + (1.74 − 1.39i)10-s + (−0.424 − 1.30i)11-s + (−1.57 − 0.512i)12-s + (1.59 + 0.519i)13-s + (0.309 + 0.951i)14-s + (−3.09 − 2.04i)15-s + (0.309 − 0.951i)16-s + (−2.21 + 3.05i)17-s + ⋯ |
| L(s) = 1 | + (0.672 − 0.218i)2-s + (−0.562 − 0.774i)3-s + (0.404 − 0.293i)4-s + (0.936 − 0.350i)5-s + (−0.547 − 0.398i)6-s + 0.377i·7-s + (0.207 − 0.286i)8-s + (0.0256 − 0.0787i)9-s + (0.553 − 0.440i)10-s + (−0.127 − 0.393i)11-s + (−0.455 − 0.147i)12-s + (0.443 + 0.144i)13-s + (0.0825 + 0.254i)14-s + (−0.798 − 0.528i)15-s + (0.0772 − 0.237i)16-s + (−0.538 + 0.740i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.149 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42113 - 1.22192i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42113 - 1.22192i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 5 | \( 1 + (-2.09 + 0.783i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + (0.974 + 1.34i)T + (-0.927 + 2.85i)T^{2} \) |
| 11 | \( 1 + (0.424 + 1.30i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.59 - 0.519i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.21 - 3.05i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (4.29 + 3.12i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-6.79 + 2.20i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (6.02 - 4.37i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.77 + 1.28i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-10.9 - 3.55i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.244 - 0.753i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 3.88iT - 43T^{2} \) |
| 47 | \( 1 + (-0.159 - 0.219i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-6.32 - 8.71i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (2.86 - 8.80i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.63 - 5.04i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (3.00 - 4.13i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-3.93 + 2.86i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-9.18 + 2.98i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (9.27 - 6.74i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (6.68 - 9.20i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (2.68 + 8.27i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.25 - 3.10i)T + (-29.9 + 92.2i)T^{2} \) |
| 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
−11.29388704648746176412844257978, −10.77387310863674196154111245363, −9.398332295476209973903097411021, −8.601797218138502058006378779908, −7.03263624703531912318311877162, −6.23143216454751127506497805458, −5.59918147268774556370365972121, −4.35045042562793560028988709120, −2.63800090407252250074953427643, −1.30484272553471390207522460490,
2.19991865439829993100524074912, 3.77661893937393573614353344440, 4.86865473059117364929198695081, 5.67211472930434763125537141363, 6.64313940636057886856797746022, 7.67256303909126098202477918798, 9.173938615727678107922684155925, 10.04153051977007080288909198044, 10.89609630618711522472565193506, 11.39309956649154772750123309695
