| L(s) = 1 | + (1.26 − 0.484i)2-s + (−2.64 − 0.138i)3-s + (−0.129 + 0.116i)4-s + (−1.91 − 1.15i)5-s + (−3.40 + 1.10i)6-s + (−1.71 − 2.01i)7-s + (−1.33 + 2.61i)8-s + (3.97 + 0.418i)9-s + (−2.97 − 0.537i)10-s + (−0.234 − 2.23i)11-s + (0.357 − 0.289i)12-s + (−4.80 + 0.761i)13-s + (−3.13 − 1.71i)14-s + (4.89 + 3.32i)15-s + (−0.378 + 3.60i)16-s + (1.49 + 2.30i)17-s + ⋯ |
| L(s) = 1 | + (0.892 − 0.342i)2-s + (−1.52 − 0.0799i)3-s + (−0.0646 + 0.0582i)4-s + (−0.855 − 0.518i)5-s + (−1.38 + 0.451i)6-s + (−0.648 − 0.761i)7-s + (−0.471 + 0.925i)8-s + (1.32 + 0.139i)9-s + (−0.940 − 0.169i)10-s + (−0.0708 − 0.673i)11-s + (0.103 − 0.0836i)12-s + (−1.33 + 0.211i)13-s + (−0.838 − 0.457i)14-s + (1.26 + 0.859i)15-s + (−0.0946 + 0.900i)16-s + (0.362 + 0.558i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0104758 - 0.273531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0104758 - 0.273531i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.91 + 1.15i)T \) |
| 7 | \( 1 + (1.71 + 2.01i)T \) |
| good | 2 | \( 1 + (-1.26 + 0.484i)T + (1.48 - 1.33i)T^{2} \) |
| 3 | \( 1 + (2.64 + 0.138i)T + (2.98 + 0.313i)T^{2} \) |
| 11 | \( 1 + (0.234 + 2.23i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (4.80 - 0.761i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.49 - 2.30i)T + (-6.91 + 15.5i)T^{2} \) |
| 19 | \( 1 + (-4.56 + 5.06i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (0.915 + 2.38i)T + (-17.0 + 15.3i)T^{2} \) |
| 29 | \( 1 + (4.81 + 1.56i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.0386 + 0.181i)T + (-28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (3.41 + 4.22i)T + (-7.69 + 36.1i)T^{2} \) |
| 41 | \( 1 + (4.30 - 5.92i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (2.22 + 2.22i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.78 + 4.40i)T + (19.1 + 42.9i)T^{2} \) |
| 53 | \( 1 + (0.117 - 2.24i)T + (-52.7 - 5.54i)T^{2} \) |
| 59 | \( 1 + (11.7 - 5.24i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-4.31 + 9.69i)T + (-40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-6.45 + 4.19i)T + (27.2 - 61.2i)T^{2} \) |
| 71 | \( 1 + (-1.99 + 6.15i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.617 - 0.500i)T + (15.1 + 71.4i)T^{2} \) |
| 79 | \( 1 + (-0.976 - 4.59i)T + (-72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-8.62 - 4.39i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-10.8 - 4.82i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (-9.62 + 4.90i)T + (57.0 - 78.4i)T^{2} \) |
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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
−12.22413092473168057431859561659, −11.55722552897639640339574808505, −10.76372825978042672669829729419, −9.409596170372794344459175350014, −7.85279383031950952562994966220, −6.71454631899316168240944882598, −5.36364886363470391368058906514, −4.64481554352018414022161015943, −3.43436616652340040613741383038, −0.22233020295838787856576304512,
3.40350342616145638411536735551, 4.88038012363668871924281027769, 5.53461434305734442736433539170, 6.62826168950846459020435079005, 7.50563485560996066015168502835, 9.605922647590956023403732472554, 10.24766168923090148963024299306, 11.79334038178944202250545115324, 12.07215198975111832086539057726, 12.85558715177409930599008364710
