| L(s) = 1 | + (−0.754 + 1.19i)2-s + (−0.139 − 0.0578i)3-s + (−0.862 − 1.80i)4-s + (2.91 − 1.48i)5-s + (0.174 − 0.123i)6-s + (−2.40 − 3.92i)7-s + (2.80 + 0.329i)8-s + (−2.10 − 2.10i)9-s + (−0.421 + 4.60i)10-s + (2.31 + 0.182i)11-s + (0.0160 + 0.301i)12-s + (0.743 + 3.09i)13-s + (6.50 + 0.0829i)14-s + (−0.492 + 0.0387i)15-s + (−2.51 + 3.11i)16-s + (0.899 − 0.768i)17-s + ⋯ |
| L(s) = 1 | + (−0.533 + 0.845i)2-s + (−0.0805 − 0.0333i)3-s + (−0.431 − 0.902i)4-s + (1.30 − 0.663i)5-s + (0.0712 − 0.0503i)6-s + (−0.908 − 1.48i)7-s + (0.993 + 0.116i)8-s + (−0.701 − 0.701i)9-s + (−0.133 + 1.45i)10-s + (0.697 + 0.0549i)11-s + (0.00462 + 0.0870i)12-s + (0.206 + 0.858i)13-s + (1.73 + 0.0221i)14-s + (−0.127 + 0.0100i)15-s + (−0.628 + 0.777i)16-s + (0.218 − 0.186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.902712 - 0.125143i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.902712 - 0.125143i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.754 - 1.19i)T \) |
| 41 | \( 1 + (-6.34 + 0.889i)T \) |
| good | 3 | \( 1 + (0.139 + 0.0578i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-2.91 + 1.48i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (2.40 + 3.92i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (-2.31 - 0.182i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (-0.743 - 3.09i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (-0.899 + 0.768i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-5.22 - 1.25i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (4.43 - 3.22i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (3.36 + 2.87i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (-1.90 - 5.87i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.773 + 2.38i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (0.766 + 0.121i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-3.64 + 5.94i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (3.29 - 3.85i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (-5.52 - 7.60i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.739 - 0.117i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-0.809 - 10.2i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (0.0829 - 1.05i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (8.87 - 8.87i)T - 73iT^{2} \) |
| 79 | \( 1 + (-4.61 + 11.1i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 7.38iT - 83T^{2} \) |
| 89 | \( 1 + (-11.7 + 7.21i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (-1.48 - 18.8i)T + (-95.8 + 15.1i)T^{2} \) |
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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
−13.24908020008613869701849582668, −11.78259312373152128645811081533, −10.28681132163851032347331385137, −9.550545476661047304385662402304, −9.006484680700157317987284389809, −7.38510288661937634344191349629, −6.41145851168156149998800083685, −5.61431896103767230973471120224, −3.96883577442254879891693010418, −1.17412611773755701216627054436,
2.26815966387438160776429384685, 3.12406027334941821002324505419, 5.46572669745849651676884007420, 6.30359723673787023245361838536, 8.035267169157253476432835681493, 9.235572753938205568474614702344, 9.757391641404562632894473859616, 10.79940757729982514637055783988, 11.79801338478931761429073423511, 12.76014523436220637751670539107
