| L(s) = 1 | + (−0.331 + 1.02i)2-s + (0.293 + 0.263i)3-s + (2.30 + 1.67i)4-s + (−0.793 + 1.37i)5-s + (−0.366 + 0.211i)6-s + (0.503 − 4.79i)7-s + (−5.94 + 4.31i)8-s + (−0.924 − 8.79i)9-s + (−1.13 − 1.26i)10-s + (−3.31 − 7.44i)11-s + (0.233 + 1.09i)12-s + (0.723 − 3.40i)13-s + (4.71 + 2.10i)14-s + (−0.595 + 0.193i)15-s + (1.08 + 3.34i)16-s + (−8.55 + 19.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.165 + 0.510i)2-s + (0.0977 + 0.0879i)3-s + (0.576 + 0.418i)4-s + (−0.158 + 0.274i)5-s + (−0.0610 + 0.0352i)6-s + (0.0719 − 0.684i)7-s + (−0.742 + 0.539i)8-s + (−0.102 − 0.977i)9-s + (−0.113 − 0.126i)10-s + (−0.301 − 0.677i)11-s + (0.0194 + 0.0916i)12-s + (0.0556 − 0.261i)13-s + (0.337 + 0.150i)14-s + (−0.0396 + 0.0128i)15-s + (0.0679 + 0.209i)16-s + (−0.503 + 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.921035 + 0.395569i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.921035 + 0.395569i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-19.3 + 24.2i)T \) |
| good | 2 | \( 1 + (0.331 - 1.02i)T + (-3.23 - 2.35i)T^{2} \) |
| 3 | \( 1 + (-0.293 - 0.263i)T + (0.940 + 8.95i)T^{2} \) |
| 5 | \( 1 + (0.793 - 1.37i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-0.503 + 4.79i)T + (-47.9 - 10.1i)T^{2} \) |
| 11 | \( 1 + (3.31 + 7.44i)T + (-80.9 + 89.9i)T^{2} \) |
| 13 | \( 1 + (-0.723 + 3.40i)T + (-154. - 68.7i)T^{2} \) |
| 17 | \( 1 + (8.55 - 19.2i)T + (-193. - 214. i)T^{2} \) |
| 19 | \( 1 + (-5.81 + 1.23i)T + (329. - 146. i)T^{2} \) |
| 23 | \( 1 + (-3.72 - 5.12i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (23.1 + 7.52i)T + (680. + 494. i)T^{2} \) |
| 37 | \( 1 + (58.1 - 33.5i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-20.8 - 23.1i)T + (-175. + 1.67e3i)T^{2} \) |
| 43 | \( 1 + (9.34 + 43.9i)T + (-1.68e3 + 752. i)T^{2} \) |
| 47 | \( 1 + (-15.6 - 48.2i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-10.4 + 1.09i)T + (2.74e3 - 584. i)T^{2} \) |
| 59 | \( 1 + (-52.3 + 58.1i)T + (-363. - 3.46e3i)T^{2} \) |
| 61 | \( 1 - 91.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (32.1 - 55.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (14.6 + 139. i)T + (-4.93e3 + 1.04e3i)T^{2} \) |
| 73 | \( 1 + (41.2 + 92.5i)T + (-3.56e3 + 3.96e3i)T^{2} \) |
| 79 | \( 1 + (13.2 - 29.8i)T + (-4.17e3 - 4.63e3i)T^{2} \) |
| 83 | \( 1 + (-57.5 + 51.8i)T + (720. - 6.85e3i)T^{2} \) |
| 89 | \( 1 + (-82.1 + 113. i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-74.1 - 53.9i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−16.90530294567250477792892862371, −15.59851486029157550268513185628, −14.82643377547792046163074389493, −13.26027801480842449280267979992, −11.79100587904423738875840373289, −10.63117147041381302612235792213, −8.777292608637559011917901799894, −7.44437531101708065153832983238, −6.13970035589132295948844727048, −3.46549600703743151666220635704,
2.35212008292846436770985879515, 5.20283385923432129194723314570, 7.11652885956764420441368366180, 8.878511358639734711750019730956, 10.31047274919594204080266966488, 11.51906180782569753179529971532, 12.58001265373362707825015706397, 14.14103952943579450612348882598, 15.51451931689855953087848935172, 16.30665140387731313449375833381
