L(s) = 1 | + (−0.766 + 0.642i)2-s + (0.203 + 0.170i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)5-s − 0.265·6-s + (1.72 − 0.628i)7-s + (0.500 + 0.866i)8-s + (−0.508 − 2.88i)9-s + (−0.5 + 0.866i)10-s + (−1.26 − 2.18i)11-s + (0.203 − 0.170i)12-s + (0.483 − 2.74i)13-s + (−0.918 + 1.59i)14-s + (0.249 + 0.0906i)15-s + (−0.939 − 0.342i)16-s + (0.786 + 4.45i)17-s + ⋯ |
L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.117 + 0.0983i)3-s + (0.0868 − 0.492i)4-s + (0.420 − 0.152i)5-s − 0.108·6-s + (0.652 − 0.237i)7-s + (0.176 + 0.306i)8-s + (−0.169 − 0.961i)9-s + (−0.158 + 0.273i)10-s + (−0.381 − 0.660i)11-s + (0.0586 − 0.0491i)12-s + (0.134 − 0.761i)13-s + (−0.245 + 0.425i)14-s + (0.0643 + 0.0234i)15-s + (−0.234 − 0.0855i)16-s + (0.190 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15987 - 0.198699i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15987 - 0.198699i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 + (-0.939 + 0.342i)T \) |
| 37 | \( 1 + (-0.599 + 6.05i)T \) |
good | 3 | \( 1 + (-0.203 - 0.170i)T + (0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (-1.72 + 0.628i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (1.26 + 2.18i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.483 + 2.74i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.786 - 4.45i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (1.19 + 1.00i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-3.64 + 6.32i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.63 - 6.30i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 41 | \( 1 + (0.798 - 4.53i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + (6.06 - 10.4i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-11.0 - 4.02i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (13.1 + 4.77i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.624 + 3.54i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (6.15 - 2.23i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (5.94 + 4.99i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 - 2.00T + 73T^{2} \) |
| 79 | \( 1 + (7.63 - 2.77i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (0.0889 + 0.504i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-7.93 - 2.88i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.127 + 0.221i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.85642449626386883827012676032, −10.59363465051563114835047123108, −9.283520678371769186672479706393, −8.583772442070591605424939668990, −7.80811468005685647695811254781, −6.50376302673587038561777042749, −5.75115393929435386743099857227, −4.51454735121760254526201447777, −2.93708923047379590311583303650, −1.04646393381960816170620820760,
1.74669675684761851819543751475, 2.73886447456201413374108909051, 4.49681327847122667169664547656, 5.49009679050595440985297895202, 7.01673731640453524133249339640, 7.78779717810108562176035522741, 8.753357537236598324318289359895, 9.680829579652933363633065929481, 10.48355506186159437847125262669, 11.44643248115803522697874192422