| L(s) = 1 | + (0.126 − 2.41i)2-s + (0.433 − 0.351i)3-s + (−3.81 − 0.400i)4-s + (1.33 − 1.79i)5-s + (−0.792 − 1.09i)6-s + (2.23 − 1.40i)7-s + (−0.693 + 4.37i)8-s + (−0.558 + 2.62i)9-s + (−4.15 − 3.45i)10-s + (−1.43 + 0.303i)11-s + (−1.79 + 1.16i)12-s + (−2.47 + 4.84i)13-s + (−3.11 − 5.57i)14-s + (−0.0481 − 1.24i)15-s + (2.97 + 0.631i)16-s + (0.526 − 0.202i)17-s + ⋯ |
| L(s) = 1 | + (0.0893 − 1.70i)2-s + (0.250 − 0.202i)3-s + (−1.90 − 0.200i)4-s + (0.598 − 0.800i)5-s + (−0.323 − 0.445i)6-s + (0.846 − 0.532i)7-s + (−0.245 + 1.54i)8-s + (−0.186 + 0.876i)9-s + (−1.31 − 1.09i)10-s + (−0.431 + 0.0916i)11-s + (−0.518 + 0.336i)12-s + (−0.685 + 1.34i)13-s + (−0.832 − 1.49i)14-s + (−0.0124 − 0.322i)15-s + (0.742 + 0.157i)16-s + (0.127 − 0.0490i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.292899 - 1.28197i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.292899 - 1.28197i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.33 + 1.79i)T \) |
| 7 | \( 1 + (-2.23 + 1.40i)T \) |
| good | 2 | \( 1 + (-0.126 + 2.41i)T + (-1.98 - 0.209i)T^{2} \) |
| 3 | \( 1 + (-0.433 + 0.351i)T + (0.623 - 2.93i)T^{2} \) |
| 11 | \( 1 + (1.43 - 0.303i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (2.47 - 4.84i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.526 + 0.202i)T + (12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (0.200 + 1.90i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-9.15 - 0.479i)T + (22.8 + 2.40i)T^{2} \) |
| 29 | \( 1 + (-2.15 + 2.96i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.68 - 3.79i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (3.86 + 5.95i)T + (-15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (-4.45 + 1.44i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.695 + 0.695i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.24 - 11.0i)T + (-34.9 - 31.4i)T^{2} \) |
| 53 | \( 1 + (1.50 + 1.85i)T + (-11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (0.683 - 0.758i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (7.98 - 7.18i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-1.92 - 5.00i)T + (-49.7 + 44.8i)T^{2} \) |
| 71 | \( 1 + (3.85 + 2.80i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (5.13 + 3.33i)T + (29.6 + 66.6i)T^{2} \) |
| 79 | \( 1 + (-0.415 - 0.934i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-9.51 - 1.50i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (6.07 + 6.74i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (7.91 - 1.25i)T + (92.2 - 29.9i)T^{2} \) |
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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
−12.27357296172469152882543004134, −11.22826119040100172229221960759, −10.57977139338888333115966717483, −9.412804334540320050641537658100, −8.695483237279755195539296952195, −7.32649055619318123284697270876, −5.07303132244827887463044940227, −4.48722096534193953460759848865, −2.55465640737212200365866944655, −1.47045732874209832037330303140,
3.04460725126248505659270611892, 4.99691255395158895341756379456, 5.76110129963575834867211492067, 6.86377309431833783058879168521, 7.86580897355303012306966418259, 8.776921324329001901753544901462, 9.801873481772652662543065820477, 10.99399280305124160039708927301, 12.50149426341757909536726156054, 13.56320834956305472358362045870
