L(s) = 1 | + (−0.165 − 0.286i)2-s + (1.34 + 2.33i)3-s + (0.945 − 1.63i)4-s + (0.445 − 0.771i)6-s + (−1.67 + 2.90i)7-s − 1.28·8-s + (−2.12 + 3.67i)9-s + (1.62 + 2.81i)11-s + 5.08·12-s + (3.39 + 1.21i)13-s + 1.10·14-s + (−1.67 − 2.90i)16-s + (0.974 − 1.68i)17-s + 1.40·18-s + (0.622 − 1.07i)19-s + ⋯ |
L(s) = 1 | + (−0.116 − 0.202i)2-s + (0.777 + 1.34i)3-s + (0.472 − 0.818i)4-s + (0.181 − 0.314i)6-s + (−0.633 + 1.09i)7-s − 0.455·8-s + (−0.707 + 1.22i)9-s + (0.489 + 0.847i)11-s + 1.46·12-s + (0.941 + 0.337i)13-s + 0.296·14-s + (−0.419 − 0.726i)16-s + (0.236 − 0.409i)17-s + 0.331·18-s + (0.142 − 0.247i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46655 + 0.774033i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46655 + 0.774033i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + (-3.39 - 1.21i)T \) |
good | 2 | \( 1 + (0.165 + 0.286i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.34 - 2.33i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (1.67 - 2.90i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.62 - 2.81i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.974 + 1.68i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.622 + 1.07i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.34 + 2.33i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.78T + 31T^{2} \) |
| 37 | \( 1 + (0.974 + 1.68i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.39 + 2.40i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.36 + 7.56i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6.86T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + (-1.26 + 2.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.74 + 6.48i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.00 - 3.47i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.62 - 4.54i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 5.46T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 8.61T + 83T^{2} \) |
| 89 | \( 1 + (5.15 + 8.93i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.63 + 4.56i)T + (-48.5 - 84.0i)T^{2} \) |
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
−11.54648978569607864815557799725, −10.60719732064113102229655767352, −9.643579598459333194349844235836, −9.365571050989130734931998969586, −8.432363825169864539162751742253, −6.75626455983017038889950930936, −5.74945805622263169193966687269, −4.60965111359895253963818418636, −3.32441812126338217459835954609, −2.18553822503122477012390471528,
1.31237083377814264592273014348, 3.06054334795429395109780906596, 3.73443216338418249785763348930, 6.21506791143344672672721428024, 6.71509159984012959426896318795, 7.79776592445192325834832770320, 8.193643156552614233279949409168, 9.272616641568434135134267704555, 10.68762201553688330072473307007, 11.64907592748408028254663983591