| L(s) = 1 | + (−0.428 + 0.247i)2-s + (−0.866 + 1.5i)3-s + (−1.87 + 3.25i)4-s + (−2.75 − 4.17i)5-s − 0.856i·6-s + (−5.82 − 3.87i)7-s − 3.83i·8-s + (−1.5 − 2.59i)9-s + (2.21 + 1.10i)10-s + (3.55 − 6.16i)11-s + (−3.25 − 5.63i)12-s − 6.94·13-s + (3.45 + 0.219i)14-s + (8.64 − 0.516i)15-s + (−6.56 − 11.3i)16-s + (−6.89 + 11.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.214 + 0.123i)2-s + (−0.288 + 0.5i)3-s + (−0.469 + 0.813i)4-s + (−0.550 − 0.834i)5-s − 0.142i·6-s + (−0.832 − 0.553i)7-s − 0.479i·8-s + (−0.166 − 0.288i)9-s + (0.221 + 0.110i)10-s + (0.323 − 0.560i)11-s + (−0.271 − 0.469i)12-s − 0.534·13-s + (0.246 + 0.0156i)14-s + (0.576 − 0.0344i)15-s + (−0.410 − 0.710i)16-s + (−0.405 + 0.702i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 + 0.602i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.798 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0222108 - 0.0663002i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0222108 - 0.0663002i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.866 - 1.5i)T \) |
| 5 | \( 1 + (2.75 + 4.17i)T \) |
| 7 | \( 1 + (5.82 + 3.87i)T \) |
| good | 2 | \( 1 + (0.428 - 0.247i)T + (2 - 3.46i)T^{2} \) |
| 11 | \( 1 + (-3.55 + 6.16i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 6.94T + 169T^{2} \) |
| 17 | \( 1 + (6.89 - 11.9i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (26.4 - 15.2i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (9.08 - 5.24i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 28.7T + 841T^{2} \) |
| 31 | \( 1 + (21.1 + 12.1i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-18.2 + 10.5i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 77.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 70.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-15.0 - 26.0i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (54.6 + 31.5i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (66.4 + 38.3i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-4.73 + 2.73i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (62.8 + 36.2i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 60.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-44.3 + 76.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-5.40 - 9.36i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 49.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + (105. - 60.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 28.1T + 9.40e3T^{2} \) |
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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
−12.79267073250602287376064705773, −12.36188944252788908701871589511, −10.96656952755999832198223071460, −9.694008007398025170788956427017, −8.755171124453773802537531950392, −7.77032582094435924949576092623, −6.26619779518780895436366401939, −4.46447038935020103405223748713, −3.64563088785009948129985071704, −0.05571040541707899805212528327,
2.48928474043447727689585439226, 4.59235660273326490669125531493, 6.20525967302493784721620893969, 7.01462777423522918263766686137, 8.623185668729197724833222208708, 9.761355613430204141426859934026, 10.73551568420690023791027764887, 11.80022654033461603179230172831, 12.81980581667056839079343936113, 13.98506175424543386564977992699
