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
−10.59364313581740396226418851353, −9.842602270186359458887246126602, −9.262537771476023361381414755058, −8.404976457794934501149056549245, −7.11177015660119844487028039779, −6.59494777955240833801046789825, −5.15353846632121513416255788334, −3.93974692621782328869453407779, −2.91727013025156978911375337054, −1.57562854729862564496882429416,
1.73851843158847415459139872757, 2.85173688046901277572882723468, 3.87278047328155861178029353761, 5.55265237679813410663185079729, 6.18627281057966197067447271898, 7.44459187551612407360104401112, 8.342726729612328186710607236677, 9.333171723822275546739914549405, 9.728052661974204243433894626917, 10.66404212400800237259833932234