| L(s) = 1 | + (−0.0825 − 1.57i)2-s + (−0.0997 − 0.0808i)3-s + (−0.483 + 0.0508i)4-s + (−1.93 − 1.11i)5-s + (−0.118 + 0.163i)6-s + (2.63 + 0.202i)7-s + (−0.373 − 2.35i)8-s + (−0.620 − 2.91i)9-s + (−1.59 + 3.14i)10-s + (−3.49 − 0.743i)11-s + (0.0523 + 0.0339i)12-s + (0.403 + 0.791i)13-s + (0.101 − 4.17i)14-s + (0.103 + 0.267i)15-s + (−4.63 + 0.984i)16-s + (6.15 + 2.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.0583 − 1.11i)2-s + (−0.0576 − 0.0466i)3-s + (−0.241 + 0.0254i)4-s + (−0.866 − 0.499i)5-s + (−0.0485 + 0.0668i)6-s + (0.997 + 0.0765i)7-s + (−0.132 − 0.833i)8-s + (−0.206 − 0.972i)9-s + (−0.505 + 0.993i)10-s + (−1.05 − 0.224i)11-s + (0.0151 + 0.00981i)12-s + (0.111 + 0.219i)13-s + (0.0270 − 1.11i)14-s + (0.0266 + 0.0691i)15-s + (−1.15 + 0.246i)16-s + (1.49 + 0.572i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.438507 - 0.971756i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.438507 - 0.971756i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.93 + 1.11i)T \) |
| 7 | \( 1 + (-2.63 - 0.202i)T \) |
| good | 2 | \( 1 + (0.0825 + 1.57i)T + (-1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (0.0997 + 0.0808i)T + (0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (3.49 + 0.743i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.403 - 0.791i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-6.15 - 2.36i)T + (12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.107 - 1.02i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-2.68 + 0.140i)T + (22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (0.242 + 0.334i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.79 - 4.02i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-5.67 + 8.74i)T + (-15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (0.884 + 0.287i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-3.39 - 3.39i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.43 - 6.33i)T + (-34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (5.17 - 6.38i)T + (-11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-3.83 - 4.25i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (3.99 + 3.59i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-3.90 + 10.1i)T + (-49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-5.23 + 3.80i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (9.62 - 6.25i)T + (29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (5.43 - 12.2i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (10.5 - 1.67i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-3.49 + 3.88i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (14.9 + 2.36i)T + (92.2 + 29.9i)T^{2} \) |
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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
−12.34224031183747493377354737373, −11.34942630966504280122967945212, −10.74362333400670285490031529673, −9.492482754637843378385169209461, −8.362224804597699335190724055186, −7.40901232207654782296715275805, −5.70782975799851445230582793175, −4.25080564090156924566601077570, −3.05991778022188021729141635215, −1.13434108344645080041028676974,
2.70784218509486283743422208273, 4.75826800579302726028119731342, 5.56501810626016505867541505376, 7.17128778178475511638814986527, 7.83477511630986241166786312720, 8.338532942096355084417170902184, 10.24147895161352968322093684481, 11.17102621183245438560379269150, 11.83707346695819399686556554786, 13.37279906005886106468150311906
