| L(s) = 1 | + (−3.41 − 1.74i)2-s + (−1.71 − 0.270i)3-s + (6.29 + 8.67i)4-s + (4.97 − 0.534i)5-s + (5.37 + 3.90i)6-s + (−0.908 + 0.908i)7-s + (−4.03 − 25.4i)8-s + (2.85 + 0.927i)9-s + (−17.9 − 6.83i)10-s + (−3.13 − 9.65i)11-s + (−8.42 − 16.5i)12-s + (12.1 − 6.16i)13-s + (4.68 − 1.52i)14-s + (−8.64 − 0.433i)15-s + (−17.3 + 53.2i)16-s + (4.75 − 0.753i)17-s + ⋯ |
| L(s) = 1 | + (−1.70 − 0.870i)2-s + (−0.570 − 0.0903i)3-s + (1.57 + 2.16i)4-s + (0.994 − 0.106i)5-s + (0.895 + 0.650i)6-s + (−0.129 + 0.129i)7-s + (−0.503 − 3.18i)8-s + (0.317 + 0.103i)9-s + (−1.79 − 0.683i)10-s + (−0.285 − 0.877i)11-s + (−0.702 − 1.37i)12-s + (0.931 − 0.474i)13-s + (0.334 − 0.108i)14-s + (−0.576 − 0.0288i)15-s + (−1.08 + 3.32i)16-s + (0.279 − 0.0442i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.119 + 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.425552 - 0.377395i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.425552 - 0.377395i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.71 + 0.270i)T \) |
| 5 | \( 1 + (-4.97 + 0.534i)T \) |
| good | 2 | \( 1 + (3.41 + 1.74i)T + (2.35 + 3.23i)T^{2} \) |
| 7 | \( 1 + (0.908 - 0.908i)T - 49iT^{2} \) |
| 11 | \( 1 + (3.13 + 9.65i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-12.1 + 6.16i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (-4.75 + 0.753i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-21.9 + 30.1i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (4.14 - 8.13i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (-14.7 - 20.3i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (25.2 + 18.3i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (12.6 + 24.8i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-1.61 + 4.97i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-44.2 - 44.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (0.220 - 1.39i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-28.1 - 4.45i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (10.4 + 3.39i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (4.02 + 12.3i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (109. - 17.3i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (40.2 - 29.2i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (33.5 - 65.9i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-16.0 - 22.0i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-5.17 - 32.6i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-9.58 + 3.11i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-19.1 + 120. i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
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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
−13.57567713255893752523372985251, −12.63331418885265054612264263295, −11.31268561758705754839375598409, −10.70846369389449670749639016943, −9.557720639585862472562102407075, −8.728034810642071058221557720248, −7.30669817844790744573932996215, −5.85562507794775045943570736562, −2.93234023168777021503668928713, −1.03321049693378587514463320551,
1.57499645950598207900718731724, 5.50872426537685627561962783754, 6.43333599585235198102885617116, 7.56788409363112531666984612535, 8.973831118465787012947743671355, 10.01918818711991170573476510940, 10.52309322789436867710466581496, 11.98997046713462792693638191683, 13.83064069593676038264955228353, 14.91168914775852998039930314858
