| L(s) = 1 | + (−0.415 − 0.909i)2-s + (−0.736 − 0.473i)3-s + (−0.654 + 0.755i)4-s + (−0.156 − 1.08i)5-s + (−0.124 + 0.866i)6-s + (−1.31 − 2.87i)7-s + (0.959 + 0.281i)8-s + (−0.927 − 2.03i)9-s + (−0.924 + 0.594i)10-s + (−0.110 − 0.768i)11-s + (0.840 − 0.246i)12-s + (−3.31 + 0.973i)13-s + (−2.06 + 2.38i)14-s + (−0.399 + 0.875i)15-s + (−0.142 − 0.989i)16-s + (5.14 + 5.93i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (−0.425 − 0.273i)3-s + (−0.327 + 0.377i)4-s + (−0.0699 − 0.486i)5-s + (−0.0508 + 0.353i)6-s + (−0.496 − 1.08i)7-s + (0.339 + 0.0996i)8-s + (−0.309 − 0.677i)9-s + (−0.292 + 0.187i)10-s + (−0.0333 − 0.231i)11-s + (0.242 − 0.0712i)12-s + (−0.919 + 0.270i)13-s + (−0.552 + 0.638i)14-s + (−0.103 + 0.226i)15-s + (−0.0355 − 0.247i)16-s + (1.24 + 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.256032 - 0.619665i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.256032 - 0.619665i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 67 | \( 1 + (8.11 + 1.03i)T \) |
| good | 3 | \( 1 + (0.736 + 0.473i)T + (1.24 + 2.72i)T^{2} \) |
| 5 | \( 1 + (0.156 + 1.08i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (1.31 + 2.87i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (0.110 + 0.768i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (3.31 - 0.973i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-5.14 - 5.93i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-2.84 + 6.22i)T + (-12.4 - 14.3i)T^{2} \) |
| 23 | \( 1 + (2.31 + 1.48i)T + (9.55 + 20.9i)T^{2} \) |
| 29 | \( 1 - 7.69T + 29T^{2} \) |
| 31 | \( 1 + (-0.776 - 0.227i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + 1.48T + 37T^{2} \) |
| 41 | \( 1 + (-1.50 - 1.73i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (1.62 + 1.87i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (-2.68 - 1.72i)T + (19.5 + 42.7i)T^{2} \) |
| 53 | \( 1 + (6.26 - 7.23i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (0.0676 + 0.0198i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.510 + 3.54i)T + (-58.5 - 17.1i)T^{2} \) |
| 71 | \( 1 + (-5.88 + 6.78i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.63 + 11.3i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (5.72 - 1.68i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (1.15 + 8.06i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (2.18 - 1.40i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−12.53904657631256055632144208221, −12.03987148858735896615137740298, −10.76878318144145564520175716844, −9.920621316428237848142746205407, −8.822866340763985791515680439106, −7.52989694101419489794024387112, −6.34912272317321298636050967310, −4.69958079433766266115593118703, −3.26612988964378586503480324131, −0.836900635556276221606974824896,
2.86455128797995243281089100862, 5.06761550202581162732485047650, 5.81203702997979945885444840988, 7.21350322703941160015753862819, 8.214203126165440561329972778485, 9.652023051231077560197540561254, 10.22045087545576012282360861569, 11.69876158233526883173915701036, 12.40727796054627982260738328558, 13.99746553506592145302222106082
