L(s) = 1 | + (−2.23 + 0.103i)5-s + (−1.10 + 0.640i)7-s + (−2.07 − 3.58i)11-s + (5.64 + 3.26i)13-s − 5.98i·17-s − 7.17·19-s + (6.52 + 3.76i)23-s + (4.97 − 0.461i)25-s + (2.59 + 4.5i)29-s + (−2.58 + 4.48i)31-s + (2.41 − 1.54i)35-s + 5.24i·37-s + (−0.340 + 0.589i)41-s + (1.10 − 0.640i)43-s + (−4.59 + 2.65i)47-s + ⋯ |
L(s) = 1 | + (−0.998 + 0.0462i)5-s + (−0.419 + 0.242i)7-s + (−0.624 − 1.08i)11-s + (1.56 + 0.904i)13-s − 1.45i·17-s − 1.64·19-s + (1.35 + 0.785i)23-s + (0.995 − 0.0923i)25-s + (0.482 + 0.835i)29-s + (−0.465 + 0.805i)31-s + (0.407 − 0.261i)35-s + 0.861i·37-s + (−0.0531 + 0.0920i)41-s + (0.169 − 0.0977i)43-s + (−0.670 + 0.387i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9856509545\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9856509545\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.23 - 0.103i)T \) |
good | 7 | \( 1 + (1.10 - 0.640i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.07 + 3.58i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-5.64 - 3.26i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 5.98iT - 17T^{2} \) |
| 19 | \( 1 + 7.17T + 19T^{2} \) |
| 23 | \( 1 + (-6.52 - 3.76i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.59 - 4.5i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.58 - 4.48i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.24iT - 37T^{2} \) |
| 41 | \( 1 + (0.340 - 0.589i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.10 + 0.640i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.59 - 2.65i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 2.21iT - 53T^{2} \) |
| 59 | \( 1 + (3.80 - 6.58i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.08 - 1.88i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-13.5 - 7.80i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 5.50T + 71T^{2} \) |
| 73 | \( 1 - 7.80iT - 73T^{2} \) |
| 79 | \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.44 - 4.87i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + (-12.4 + 7.16i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.271350664283207670350333202708, −8.706561163403532736923246490451, −8.171715985398193649275658609456, −7.00701649139591392462931171729, −6.52488218892787160611147635753, −5.43583163860266732619649619967, −4.50217200387755141786280460209, −3.48643879833315365314902242278, −2.84869236868344466891175093333, −1.07773322914300897378134649054,
0.45684484790436249001740010732, 2.08500923067404500357886931116, 3.42457915388335305270823009388, 4.06178199415867669838938512672, 4.95408475182590476621859244085, 6.18834234955227352240568042187, 6.74045510314424866455485732960, 7.893373386942780173730973863917, 8.252729714053018793997255512194, 9.067049503810935870094232130370