L(s) = 1 | + (0.690 − 1.58i)3-s + (1.10 − 1.91i)5-s + (0.234 − 2.63i)7-s + (−2.04 − 2.19i)9-s + (0.157 − 0.0909i)11-s + (−2.50 + 1.44i)13-s + (−2.27 − 3.07i)15-s + (−1.98 + 3.43i)17-s + (0.867 − 0.500i)19-s + (−4.02 − 2.19i)21-s + (4.86 + 2.80i)23-s + (0.0605 + 0.104i)25-s + (−4.89 + 1.73i)27-s + (0.703 + 0.406i)29-s − 7.96i·31-s + ⋯ |
L(s) = 1 | + (0.398 − 0.917i)3-s + (0.493 − 0.855i)5-s + (0.0885 − 0.996i)7-s + (−0.682 − 0.731i)9-s + (0.0475 − 0.0274i)11-s + (−0.694 + 0.400i)13-s + (−0.587 − 0.793i)15-s + (−0.481 + 0.833i)17-s + (0.198 − 0.114i)19-s + (−0.878 − 0.478i)21-s + (1.01 + 0.585i)23-s + (0.0121 + 0.0209i)25-s + (−0.942 + 0.334i)27-s + (0.130 + 0.0754i)29-s − 1.43i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.874232 - 1.36690i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.874232 - 1.36690i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.690 + 1.58i)T \) |
| 7 | \( 1 + (-0.234 + 2.63i)T \) |
good | 5 | \( 1 + (-1.10 + 1.91i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.157 + 0.0909i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.50 - 1.44i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.98 - 3.43i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.867 + 0.500i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.86 - 2.80i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.703 - 0.406i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7.96iT - 31T^{2} \) |
| 37 | \( 1 + (-1.25 - 2.17i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.612 - 1.06i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.47 + 9.48i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 7.15T + 47T^{2} \) |
| 53 | \( 1 + (-1.75 - 1.01i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 6.55T + 59T^{2} \) |
| 61 | \( 1 + 8.05iT - 61T^{2} \) |
| 67 | \( 1 - 6.89T + 67T^{2} \) |
| 71 | \( 1 - 11.4iT - 71T^{2} \) |
| 73 | \( 1 + (10.1 + 5.88i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + (-7.19 + 12.4i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.11 - 12.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.01 - 1.73i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.66974224911277700079438222806, −9.485492958388062049441734389902, −8.909118690160930543945997725969, −7.81649144031480346027659292206, −7.12725075649035508054791570676, −6.11613808808540871217171207974, −4.95234116201448246823153985529, −3.76212974701197599077163092029, −2.18372313336380918215096517800, −0.963718531219959311132149874758,
2.47823561961587200408961098438, 3.01832664645336189473233375098, 4.63542299078419337352462939594, 5.45221188066457406472783814473, 6.54555621717675846884865961625, 7.66318070616358753102258584959, 8.838126216931429930630905151374, 9.377138262486188276781749775620, 10.33559847497111015430773765877, 10.94600881953066158767646993054