L(s) = 1 | + (0.5 − 0.866i)2-s + (2.13 − 1.23i)3-s + (−0.499 − 0.866i)4-s + (0.511 − 0.885i)5-s − 2.46i·6-s + (−2.61 + 0.396i)7-s − 0.999·8-s + (1.53 − 2.66i)9-s + (−0.511 − 0.885i)10-s + (1.28 − 0.742i)11-s + (−2.13 − 1.23i)12-s − 0.0815i·13-s + (−0.964 + 2.46i)14-s − 2.52i·15-s + (−0.5 + 0.866i)16-s + (1.20 + 2.08i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (1.23 − 0.711i)3-s + (−0.249 − 0.433i)4-s + (0.228 − 0.395i)5-s − 1.00i·6-s + (−0.988 + 0.149i)7-s − 0.353·8-s + (0.513 − 0.888i)9-s + (−0.161 − 0.279i)10-s + (0.387 − 0.224i)11-s + (−0.616 − 0.355i)12-s − 0.0226i·13-s + (−0.257 + 0.658i)14-s − 0.650i·15-s + (−0.125 + 0.216i)16-s + (0.292 + 0.506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36463 - 1.60307i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36463 - 1.60307i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.61 - 0.396i)T \) |
| 23 | \( 1 + (-3.64 - 3.11i)T \) |
good | 3 | \( 1 + (-2.13 + 1.23i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.511 + 0.885i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.28 + 0.742i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 0.0815iT - 13T^{2} \) |
| 17 | \( 1 + (-1.20 - 2.08i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.03 + 1.79i)T + (-9.5 - 16.4i)T^{2} \) |
| 29 | \( 1 + 0.342T + 29T^{2} \) |
| 31 | \( 1 + (2.67 - 1.54i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.135 - 0.0780i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.13iT - 41T^{2} \) |
| 43 | \( 1 + 6.69iT - 43T^{2} \) |
| 47 | \( 1 + (3.72 + 2.15i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.03 - 4.06i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.48 + 1.43i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.09 + 1.89i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (13.5 - 7.84i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.61T + 71T^{2} \) |
| 73 | \( 1 + (-5.53 + 3.19i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.55 - 3.78i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.61T + 83T^{2} \) |
| 89 | \( 1 + (-7.27 + 12.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−11.58405781305097634665569828456, −10.35262646501085807297417080649, −9.199388533035740438775559232962, −8.925072334332176805712594358243, −7.59512441305498137722878539925, −6.56476866786000613217973308771, −5.31985541670502231119058091718, −3.66108181233524140662744112991, −2.86789035443006757047311345705, −1.47184863354212302951381440263,
2.73784565340108360657949106252, 3.55166782150826415158097849412, 4.65068919643822555307878136266, 6.14016736996820700841466627441, 7.06033213560197283791653825827, 8.123952791543737634865472190763, 9.178201127826243049607852626108, 9.687983240402802412499388941481, 10.65746720727275653098738772703, 12.12005732126500861101076630724