| L(s) = 1 | + (−0.207 − 0.978i)2-s + (0.796 − 2.45i)3-s + (−0.913 + 0.406i)4-s + (0.0313 − 0.297i)5-s + (−2.56 − 0.269i)6-s + (0.354 + 0.319i)7-s + (0.587 + 0.809i)8-s + (−2.94 − 2.14i)9-s + (−0.297 + 0.0313i)10-s + 3.11i·11-s + (0.269 + 2.56i)12-s + (−0.341 − 0.592i)13-s + (0.238 − 0.413i)14-s + (−0.705 − 0.314i)15-s + (0.669 − 0.743i)16-s + (−2.02 − 4.54i)17-s + ⋯ |
| L(s) = 1 | + (−0.147 − 0.691i)2-s + (0.459 − 1.41i)3-s + (−0.456 + 0.203i)4-s + (0.0140 − 0.133i)5-s + (−1.04 − 0.110i)6-s + (0.134 + 0.120i)7-s + (0.207 + 0.286i)8-s + (−0.982 − 0.714i)9-s + (−0.0942 + 0.00990i)10-s + 0.938i·11-s + (0.0777 + 0.740i)12-s + (−0.0948 − 0.164i)13-s + (0.0637 − 0.110i)14-s + (−0.182 − 0.0811i)15-s + (0.167 − 0.185i)16-s + (−0.490 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.431 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.577965 - 0.917433i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.577965 - 0.917433i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.207 + 0.978i)T \) |
| 61 | \( 1 + (6.30 - 4.61i)T \) |
| good | 3 | \( 1 + (-0.796 + 2.45i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (-0.0313 + 0.297i)T + (-4.89 - 1.03i)T^{2} \) |
| 7 | \( 1 + (-0.354 - 0.319i)T + (0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 - 3.11iT - 11T^{2} \) |
| 13 | \( 1 + (0.341 + 0.592i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.02 + 4.54i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-4.61 - 5.12i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-2.44 + 3.36i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (3.75 + 2.16i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.02 - 9.53i)T + (-28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (3.12 - 1.01i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.86 - 5.72i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.609 + 1.37i)T + (-28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (4.64 - 8.04i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.23 + 3.07i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (2.72 + 12.8i)T + (-53.8 + 23.9i)T^{2} \) |
| 67 | \( 1 + (5.27 + 0.554i)T + (65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (-2.38 + 0.250i)T + (69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-0.740 - 7.04i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (4.09 - 9.20i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (13.7 - 2.92i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (2.24 + 0.729i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-10.6 - 2.26i)T + (88.6 + 39.4i)T^{2} \) |
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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
−12.81331707491937943909899015112, −12.37359647848265263248385257569, −11.32919521330074968661317021616, −9.904588687444077967562407781511, −8.797742395764172911616832200322, −7.70594978115212284530822700800, −6.82068200907754380876958719274, −4.98999361155900229451877544820, −2.95064416277591477882270588318, −1.53039052801096860800021250905,
3.34063516033890874075406422085, 4.58634363737557118783996494198, 5.79576190099793875660347381069, 7.36916722274030950991815772718, 8.751486669995271215314876932163, 9.304208549166747696215498850635, 10.52860231992589971493181648172, 11.29335824420093447862033919338, 13.15528577931225471497890934160, 14.08080837357270076383801955616
