L(s) = 1 | + (−0.221 − 1.39i)2-s + (−1.90 + 0.618i)4-s + (−3.93 − 2.00i)5-s + (1.28 + 2.52i)8-s + (−2.12 + 2.12i)9-s + (−1.92 + 5.93i)10-s + (1.04 − 4.33i)13-s + (3.23 − 2.35i)16-s + (−6.27 − 5.35i)17-s + (3.43 + 2.49i)18-s + (8.71 + 1.38i)20-s + (8.50 + 11.7i)25-s + (−6.28 − 0.494i)26-s + (5.72 − 4.88i)29-s + (−4 − 4i)32-s + ⋯ |
L(s) = 1 | + (−0.156 − 0.987i)2-s + (−0.951 + 0.309i)4-s + (−1.75 − 0.895i)5-s + (0.453 + 0.891i)8-s + (−0.707 + 0.707i)9-s + (−0.609 + 1.87i)10-s + (0.288 − 1.20i)13-s + (0.809 − 0.587i)16-s + (−1.52 − 1.29i)17-s + (0.809 + 0.587i)18-s + (1.94 + 0.308i)20-s + (1.70 + 2.34i)25-s + (−1.23 − 0.0970i)26-s + (1.06 − 0.907i)29-s + (−0.707 − 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0633156 + 0.358530i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0633156 + 0.358530i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.221 + 1.39i)T \) |
| 41 | \( 1 + (6.39 - 0.297i)T \) |
good | 3 | \( 1 + (2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (3.93 + 2.00i)T + (2.93 + 4.04i)T^{2} \) |
| 7 | \( 1 + (-3.17 - 6.23i)T^{2} \) |
| 11 | \( 1 + (10.8 - 1.72i)T^{2} \) |
| 13 | \( 1 + (-1.04 + 4.33i)T + (-11.5 - 5.90i)T^{2} \) |
| 17 | \( 1 + (6.27 + 5.35i)T + (2.65 + 16.7i)T^{2} \) |
| 19 | \( 1 + (16.9 - 8.62i)T^{2} \) |
| 23 | \( 1 + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-5.72 + 4.88i)T + (4.53 - 28.6i)T^{2} \) |
| 31 | \( 1 + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (2.23 + 6.87i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (-21.3 + 41.8i)T^{2} \) |
| 53 | \( 1 + (-2.53 - 2.96i)T + (-8.29 + 52.3i)T^{2} \) |
| 59 | \( 1 + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (6.07 + 0.962i)T + (58.0 + 18.8i)T^{2} \) |
| 67 | \( 1 + (-66.1 - 10.4i)T^{2} \) |
| 71 | \( 1 + (-70.1 + 11.1i)T^{2} \) |
| 73 | \( 1 + (0.538 + 0.538i)T + 73iT^{2} \) |
| 79 | \( 1 + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + (2.65 + 1.62i)T + (40.4 + 79.2i)T^{2} \) |
| 97 | \( 1 + (-0.439 + 5.58i)T + (-95.8 - 15.1i)T^{2} \) |
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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
−12.07757922073379447545086701594, −11.38526596747991724990534682263, −10.65701004362550435989886743819, −9.039244727456703176893924255886, −8.349532829359873826196879038085, −7.54312094962184118508489956597, −5.17135252001415058671581780640, −4.28606487650745215684607153785, −2.90753151732982403958588325650, −0.36324009055790117107441458093,
3.54542777908827734919233828216, 4.48245395875141134538686471084, 6.48073644870006459075885606485, 6.89715969552752269899718670369, 8.319463634678402802743884907957, 8.784269035080205796500801546516, 10.46761560052722854314728518960, 11.41538310797861849622826540096, 12.28714370669162747937898572410, 13.71587610208514878201719934880