| L(s) = 1 | + (0.897 − 2.76i)2-s + (−0.144 + 0.678i)3-s + (−3.59 − 2.61i)4-s + (0.894 + 1.54i)5-s + (1.74 + 1.00i)6-s + (−8.04 + 3.58i)7-s + (−1.03 + 0.754i)8-s + (7.78 + 3.46i)9-s + (5.08 − 1.08i)10-s + (−0.441 − 0.0463i)11-s + (2.28 − 2.06i)12-s + (1.07 + 0.970i)13-s + (2.67 + 25.4i)14-s + (−1.17 + 0.383i)15-s + (−4.33 − 13.3i)16-s + (−24.1 + 2.53i)17-s + ⋯ |
| L(s) = 1 | + (0.448 − 1.38i)2-s + (−0.0480 + 0.226i)3-s + (−0.898 − 0.652i)4-s + (0.178 + 0.309i)5-s + (0.290 + 0.167i)6-s + (−1.14 + 0.511i)7-s + (−0.129 + 0.0943i)8-s + (0.864 + 0.385i)9-s + (0.508 − 0.108i)10-s + (−0.0401 − 0.00421i)11-s + (0.190 − 0.171i)12-s + (0.0829 + 0.0746i)13-s + (0.191 + 1.81i)14-s + (−0.0786 + 0.0255i)15-s + (−0.271 − 0.834i)16-s + (−1.41 + 0.149i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.920781 - 0.702639i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.920781 - 0.702639i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (3.30 + 30.8i)T \) |
| good | 2 | \( 1 + (-0.897 + 2.76i)T + (-3.23 - 2.35i)T^{2} \) |
| 3 | \( 1 + (0.144 - 0.678i)T + (-8.22 - 3.66i)T^{2} \) |
| 5 | \( 1 + (-0.894 - 1.54i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (8.04 - 3.58i)T + (32.7 - 36.4i)T^{2} \) |
| 11 | \( 1 + (0.441 + 0.0463i)T + (118. + 25.1i)T^{2} \) |
| 13 | \( 1 + (-1.07 - 0.970i)T + (17.6 + 168. i)T^{2} \) |
| 17 | \( 1 + (24.1 - 2.53i)T + (282. - 60.0i)T^{2} \) |
| 19 | \( 1 + (18.1 + 20.1i)T + (-37.7 + 359. i)T^{2} \) |
| 23 | \( 1 + (-16.7 - 23.0i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (-27.6 - 8.98i)T + (680. + 494. i)T^{2} \) |
| 37 | \( 1 + (-38.1 - 21.9i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-37.1 + 7.89i)T + (1.53e3 - 683. i)T^{2} \) |
| 43 | \( 1 + (35.7 - 32.1i)T + (193. - 1.83e3i)T^{2} \) |
| 47 | \( 1 + (-2.80 - 8.64i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-24.7 + 55.6i)T + (-1.87e3 - 2.08e3i)T^{2} \) |
| 59 | \( 1 + (-44.8 - 9.53i)T + (3.18e3 + 1.41e3i)T^{2} \) |
| 61 | \( 1 - 48.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (27.8 + 48.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (44.4 + 19.7i)T + (3.37e3 + 3.74e3i)T^{2} \) |
| 73 | \( 1 + (123. + 13.0i)T + (5.21e3 + 1.10e3i)T^{2} \) |
| 79 | \( 1 + (9.62 - 1.01i)T + (6.10e3 - 1.29e3i)T^{2} \) |
| 83 | \( 1 + (-33.8 - 159. i)T + (-6.29e3 + 2.80e3i)T^{2} \) |
| 89 | \( 1 + (-20.4 + 28.1i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (57.8 + 42.0i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−16.27956078938511372614822921191, −15.21897024058388759803946928062, −13.30880286342609086893379107408, −12.91946754648492598427858224518, −11.38217249361636961100336371151, −10.33462422693902171024456661501, −9.246443159364876343574525636621, −6.69416954347213615806766674847, −4.43574766856116221669115785700, −2.62621381069270682369314198077,
4.34981309575060166026282370241, 6.28167918028866353458700939018, 7.08924280443015149611747179277, 8.811584552542847533920409987139, 10.43975621680014766516465674083, 12.74286878970174557030573984702, 13.31612554664724100217217616085, 14.74467407747333062294946971782, 15.85936862145006322332912496706, 16.59651033482787545807105076915
