L(s) = 1 | + (1.72 − 0.181i)3-s + 2.10·5-s + (−1.78 + 1.95i)7-s + (2.93 − 0.625i)9-s + 0.399·11-s + (1.44 − 2.49i)13-s + (3.62 − 0.381i)15-s + (−0.176 + 0.305i)17-s + (2.84 + 4.93i)19-s + (−2.71 + 3.68i)21-s − 0.877·23-s − 0.571·25-s + (4.94 − 1.60i)27-s + (0.874 + 1.51i)29-s + (−4.56 − 7.91i)31-s + ⋯ |
L(s) = 1 | + (0.994 − 0.104i)3-s + 0.941·5-s + (−0.674 + 0.738i)7-s + (0.978 − 0.208i)9-s + 0.120·11-s + (0.400 − 0.693i)13-s + (0.935 − 0.0985i)15-s + (−0.0428 + 0.0741i)17-s + (0.653 + 1.13i)19-s + (−0.593 + 0.804i)21-s − 0.182·23-s − 0.114·25-s + (0.950 − 0.309i)27-s + (0.162 + 0.281i)29-s + (−0.820 − 1.42i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.20743 + 0.115165i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.20743 + 0.115165i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.72 + 0.181i)T \) |
| 7 | \( 1 + (1.78 - 1.95i)T \) |
good | 5 | \( 1 - 2.10T + 5T^{2} \) |
| 11 | \( 1 - 0.399T + 11T^{2} \) |
| 13 | \( 1 + (-1.44 + 2.49i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.176 - 0.305i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.84 - 4.93i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 0.877T + 23T^{2} \) |
| 29 | \( 1 + (-0.874 - 1.51i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.56 + 7.91i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.39 + 5.88i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.20 + 2.08i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.276 - 0.479i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.86 - 10.1i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.07 - 3.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.66 - 8.07i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.03 + 8.72i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.601 + 1.04i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + (-0.315 + 0.546i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.24 + 2.15i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.59 + 7.95i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.29 - 12.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.84 + 13.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.66404212400800237259833932234, −9.728052661974204243433894626917, −9.333171723822275546739914549405, −8.342726729612328186710607236677, −7.44459187551612407360104401112, −6.18627281057966197067447271898, −5.55265237679813410663185079729, −3.87278047328155861178029353761, −2.85173688046901277572882723468, −1.73851843158847415459139872757,
1.57562854729862564496882429416, 2.91727013025156978911375337054, 3.93974692621782328869453407779, 5.15353846632121513416255788334, 6.59494777955240833801046789825, 7.11177015660119844487028039779, 8.404976457794934501149056549245, 9.262537771476023361381414755058, 9.842602270186359458887246126602, 10.59364313581740396226418851353