L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.508 − 0.293i)3-s + (−0.499 + 0.866i)4-s + (0.863 + 1.49i)5-s + 0.587i·6-s + (−1.63 − 2.08i)7-s + 0.999·8-s + (−1.32 − 2.29i)9-s + (0.863 − 1.49i)10-s + (3.46 + 2.00i)11-s + (0.508 − 0.293i)12-s − 5.60i·13-s + (−0.987 + 2.45i)14-s − 1.01i·15-s + (−0.5 − 0.866i)16-s + (2.91 − 5.05i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.293 − 0.169i)3-s + (−0.249 + 0.433i)4-s + (0.386 + 0.668i)5-s + 0.239i·6-s + (−0.616 − 0.787i)7-s + 0.353·8-s + (−0.442 − 0.766i)9-s + (0.273 − 0.472i)10-s + (1.04 + 0.603i)11-s + (0.146 − 0.0848i)12-s − 1.55i·13-s + (−0.263 + 0.656i)14-s − 0.262i·15-s + (−0.125 − 0.216i)16-s + (0.708 − 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.581471 - 0.709095i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.581471 - 0.709095i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (1.63 + 2.08i)T \) |
| 23 | \( 1 + (-2.15 + 4.28i)T \) |
good | 3 | \( 1 + (0.508 + 0.293i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.863 - 1.49i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.46 - 2.00i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.60iT - 13T^{2} \) |
| 17 | \( 1 + (-2.91 + 5.05i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.863 - 1.49i)T + (-9.5 + 16.4i)T^{2} \) |
| 29 | \( 1 + 9.71T + 29T^{2} \) |
| 31 | \( 1 + (-3.47 - 2.00i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.43 - 4.29i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 5.60iT - 41T^{2} \) |
| 43 | \( 1 - 3.11iT - 43T^{2} \) |
| 47 | \( 1 + (-6.35 + 3.66i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.09 - 1.20i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.45 - 3.14i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.19 - 2.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.25 - 5.34i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + (-3.47 - 2.00i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.75 - 5.05i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + (7.80 + 13.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−11.31576332898212879894391151309, −10.28563576268643415464445798008, −9.828539227342183361482929380353, −8.774537573309437155439204756813, −7.34743476737733136820603046394, −6.69618520400456636093793681068, −5.46762955438429436628146421376, −3.76069308529121319011823582678, −2.86046675969596192243155773858, −0.816065964151386762035711075088,
1.75106627586566669586479221598, 3.82103371482868153081493568975, 5.29566868964345422971860218254, 5.91205889060242267469186279238, 6.93783270515988480350331573037, 8.325264141785479935326996653557, 9.138274596109313523370146007471, 9.589163654547954633693070729770, 11.01848668064269117190364467453, 11.75232589799705725419348319113