L(s) = 1 | + (−0.203 + 0.280i)2-s + (−1.27 + 2.87i)3-s + (0.580 + 1.78i)4-s + (2.15 + 0.608i)5-s + (−0.544 − 0.943i)6-s + (−2.13 − 1.91i)7-s + (−1.27 − 0.415i)8-s + (−4.60 − 5.10i)9-s + (−0.609 + 0.479i)10-s + (1.68 + 0.358i)11-s + (−5.87 − 0.617i)12-s + (5.44 − 0.571i)13-s + (0.972 − 0.206i)14-s + (−4.49 + 5.39i)15-s + (−2.66 + 1.93i)16-s + (0.244 + 1.15i)17-s + ⋯ |
L(s) = 1 | + (−0.144 + 0.198i)2-s + (−0.737 + 1.65i)3-s + (0.290 + 0.893i)4-s + (0.962 + 0.272i)5-s + (−0.222 − 0.385i)6-s + (−0.805 − 0.725i)7-s + (−0.452 − 0.146i)8-s + (−1.53 − 1.70i)9-s + (−0.192 + 0.151i)10-s + (0.508 + 0.108i)11-s + (−1.69 − 0.178i)12-s + (1.50 − 0.158i)13-s + (0.259 − 0.0552i)14-s + (−1.16 + 1.39i)15-s + (−0.666 + 0.483i)16-s + (0.0594 + 0.279i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.337679 + 0.876162i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.337679 + 0.876162i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.15 - 0.608i)T \) |
| 31 | \( 1 + (-5.43 + 1.21i)T \) |
good | 2 | \( 1 + (0.203 - 0.280i)T + (-0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (1.27 - 2.87i)T + (-2.00 - 2.22i)T^{2} \) |
| 7 | \( 1 + (2.13 + 1.91i)T + (0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-1.68 - 0.358i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-5.44 + 0.571i)T + (12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-0.244 - 1.15i)T + (-15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.251 - 2.39i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (6.26 + 2.03i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.45 - 1.06i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (-0.390 + 0.225i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.11 - 1.83i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-3.70 - 0.389i)T + (42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-2.94 - 4.05i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.06 + 1.86i)T + (5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (12.4 + 5.55i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 - 8.17T + 61T^{2} \) |
| 67 | \( 1 + (-8.49 - 4.90i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.09 + 2.33i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-0.448 + 2.11i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-9.59 + 2.04i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (2.37 + 5.33i)T + (-55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (5.21 + 16.0i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (5.66 - 1.84i)T + (78.4 - 57.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.36241729477154743154798150790, −12.15714386998780038201073694250, −11.04181030424042447395865125920, −10.28373905399837846016391247996, −9.523024207005962477001119836724, −8.435769117051605036189942568044, −6.52047292632219825027838376715, −5.98144891451591349554660999722, −4.17450492034715461880428146120, −3.35029087752451882908204799272,
1.15706793654418470935796642845, 2.38164081974799599582861671688, 5.52012645524742205942978649628, 6.18741334789453263461063697863, 6.69388289736618356350974720615, 8.487609740114407932899904846277, 9.515821463203165570848731416818, 10.75358521270671830006144823133, 11.74089348392151393879079742298, 12.46236377049040801534036574585