| L(s) = 1 | + (0.818 + 1.15i)2-s + (0.729 + 0.0637i)3-s + (−0.661 + 1.88i)4-s + (1.85 − 0.865i)5-s + (0.522 + 0.893i)6-s + (0.692 + 0.399i)7-s + (−2.71 + 0.781i)8-s + (−2.42 − 0.427i)9-s + (2.51 + 1.43i)10-s + (4.45 + 1.19i)11-s + (−0.602 + 1.33i)12-s + (0.228 + 2.61i)13-s + (0.105 + 1.12i)14-s + (1.40 − 0.512i)15-s + (−3.12 − 2.49i)16-s + (0.295 + 1.67i)17-s + ⋯ |
| L(s) = 1 | + (0.578 + 0.815i)2-s + (0.420 + 0.0368i)3-s + (−0.330 + 0.943i)4-s + (0.830 − 0.387i)5-s + (0.213 + 0.364i)6-s + (0.261 + 0.151i)7-s + (−0.961 + 0.276i)8-s + (−0.808 − 0.142i)9-s + (0.796 + 0.453i)10-s + (1.34 + 0.360i)11-s + (−0.173 + 0.385i)12-s + (0.0634 + 0.725i)13-s + (0.0281 + 0.300i)14-s + (0.363 − 0.132i)15-s + (−0.781 − 0.623i)16-s + (0.0716 + 0.406i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.65898 + 1.26579i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65898 + 1.26579i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.818 - 1.15i)T \) |
| 19 | \( 1 + (0.748 + 4.29i)T \) |
| good | 3 | \( 1 + (-0.729 - 0.0637i)T + (2.95 + 0.520i)T^{2} \) |
| 5 | \( 1 + (-1.85 + 0.865i)T + (3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (-0.692 - 0.399i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.45 - 1.19i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.228 - 2.61i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.295 - 1.67i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (2.72 + 7.48i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (2.69 - 1.88i)T + (9.91 - 27.2i)T^{2} \) |
| 31 | \( 1 + (-0.290 + 0.502i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.32 + 2.32i)T - 37iT^{2} \) |
| 41 | \( 1 + (-5.98 - 7.13i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.98 - 1.85i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (-1.42 + 8.09i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.43 + 7.36i)T + (-34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (4.02 + 2.82i)T + (20.1 + 55.4i)T^{2} \) |
| 61 | \( 1 + (6.41 + 2.98i)T + (39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (-1.01 + 0.709i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (2.51 - 6.92i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.15 - 4.94i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-12.6 + 10.6i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (3.72 - 0.997i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (8.21 - 9.78i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.38 - 13.5i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.07563367757405483426372545101, −11.27879807796319019105210646630, −9.552384650940220331306621157573, −8.973262480334151822722230593110, −8.208516850025307150971362885859, −6.76522775104714284343648054910, −6.09739153078828429375737634860, −4.91093811623452432659327114168, −3.82841352483186133879939781796, −2.23441974637541688607733159748,
1.63096023056830203575138618380, 2.95458565800172523724096055991, 3.99522014821640733109233021119, 5.63446958281931766592302540977, 6.13245320038766980776516251310, 7.77420492985054381515503333006, 9.054762488315832847910711731270, 9.709443941424574031791018264786, 10.72819784238461090942080370628, 11.53744770740078646967487350980
