| L(s) = 1 | + (−0.996 − 0.0871i)2-s + (0.984 + 0.173i)4-s + (2.00 − 0.997i)5-s + (2.07 + 1.45i)7-s + (−0.965 − 0.258i)8-s + (−2.08 + 0.819i)10-s + (1.56 − 4.29i)11-s + (0.388 + 4.43i)13-s + (−1.94 − 1.62i)14-s + (0.939 + 0.342i)16-s + (3.33 − 0.892i)17-s + (0.649 − 0.374i)19-s + (2.14 − 0.634i)20-s + (−1.93 + 4.14i)22-s + (0.184 + 0.262i)23-s + ⋯ |
| L(s) = 1 | + (−0.704 − 0.0616i)2-s + (0.492 + 0.0868i)4-s + (0.894 − 0.446i)5-s + (0.784 + 0.549i)7-s + (−0.341 − 0.0915i)8-s + (−0.657 + 0.259i)10-s + (0.471 − 1.29i)11-s + (0.107 + 1.23i)13-s + (−0.518 − 0.435i)14-s + (0.234 + 0.0855i)16-s + (0.808 − 0.216i)17-s + (0.148 − 0.0860i)19-s + (0.479 − 0.141i)20-s + (−0.411 + 0.883i)22-s + (0.0383 + 0.0548i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.49449 - 0.171399i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49449 - 0.171399i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.996 + 0.0871i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.00 + 0.997i)T \) |
| good | 7 | \( 1 + (-2.07 - 1.45i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-1.56 + 4.29i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.388 - 4.43i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-3.33 + 0.892i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.649 + 0.374i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.184 - 0.262i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (7.59 - 6.37i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.402 + 2.28i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.33 + 8.71i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (5.59 - 6.66i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.41 - 11.6i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-2.83 + 4.05i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-6.98 + 6.98i)T - 53iT^{2} \) |
| 59 | \( 1 + (-4.25 + 1.54i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.193 + 1.09i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.09 + 0.708i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (1.12 + 0.648i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.981 + 3.66i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.09 - 3.69i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.504 - 5.76i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (3.81 + 6.61i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.43 + 1.13i)T + (62.3 - 74.3i)T^{2} \) |
| 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
−10.03164796067776539793940017006, −9.144368127058983598526491084306, −8.822345075564137592287020352987, −7.890094132197462494162754052119, −6.74922949178136512751268104472, −5.83354643932156645600495911612, −5.11066526754957193845520720340, −3.63320492665056300324768294794, −2.19644656559832377781580974882, −1.21793654884269146654006178844,
1.28734255822379099854090127110, 2.33572400175822501405385321180, 3.74916879384078993393437722013, 5.13922265384643165778565488969, 5.94187137728089201707780135020, 7.12913448724357253964937602325, 7.58716522304091753054025888662, 8.611968650672615893232411237542, 9.631961640267117291284018628455, 10.23613673942516901422687227556
