| L(s) = 1 | + (0.670 + 0.0351i)2-s + (1.37 + 1.70i)3-s + (−1.54 − 0.161i)4-s + (−0.425 + 2.19i)5-s + (0.864 + 1.19i)6-s + (2.46 − 0.965i)7-s + (−2.35 − 0.372i)8-s + (−0.373 + 1.75i)9-s + (−0.362 + 1.45i)10-s + (−2.32 + 0.495i)11-s + (−1.84 − 2.84i)12-s + (4.81 + 2.45i)13-s + (1.68 − 0.561i)14-s + (−4.32 + 2.30i)15-s + (1.46 + 0.310i)16-s + (−1.97 − 5.13i)17-s + ⋯ |
| L(s) = 1 | + (0.474 + 0.0248i)2-s + (0.795 + 0.982i)3-s + (−0.770 − 0.0809i)4-s + (−0.190 + 0.981i)5-s + (0.353 + 0.486i)6-s + (0.931 − 0.364i)7-s + (−0.832 − 0.131i)8-s + (−0.124 + 0.585i)9-s + (−0.114 + 0.461i)10-s + (−0.702 + 0.149i)11-s + (−0.533 − 0.821i)12-s + (1.33 + 0.680i)13-s + (0.450 − 0.149i)14-s + (−1.11 + 0.594i)15-s + (0.365 + 0.0777i)16-s + (−0.477 − 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30280 + 0.855420i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30280 + 0.855420i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.425 - 2.19i)T \) |
| 7 | \( 1 + (-2.46 + 0.965i)T \) |
| good | 2 | \( 1 + (-0.670 - 0.0351i)T + (1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (-1.37 - 1.70i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (2.32 - 0.495i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-4.81 - 2.45i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (1.97 + 5.13i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.0661 + 0.629i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.210 + 4.01i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-3.54 + 4.87i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.355 - 0.798i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (3.37 - 2.19i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (4.20 - 1.36i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-4.81 - 4.81i)T + 43iT^{2} \) |
| 47 | \( 1 + (9.20 + 3.53i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (8.11 - 6.56i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-2.26 + 2.52i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-1.38 + 1.24i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (8.85 - 3.40i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (5.22 + 3.79i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.92 - 7.58i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (-4.19 - 9.43i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-1.39 + 8.79i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-9.71 - 10.7i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-1.14 - 7.25i)T + (-92.2 + 29.9i)T^{2} \) |
| 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
−13.42129245140195514291496156625, −11.75872811445396132755838099074, −10.80696870726868930435246140138, −9.917044098016285672275691683898, −8.872786479515382748557497502817, −8.028838906779965608471243658153, −6.50664802309872302938632821814, −4.84525800419017450036817828712, −4.07802645734576040956432028829, −2.90912273514645108724111729240,
1.55952053267046213980532458922, 3.47085088242276086379882056884, 4.87986159381770206398288776521, 5.88909000568283440532908838549, 7.81940501988837893647127398804, 8.420002472809857540004544556168, 8.907913073636682438339908050921, 10.67293653131460947067444788221, 12.00552440078772885717517343321, 12.95729068449362780236124977188
