L(s) = 1 | + (0.0274 − 1.41i)2-s + (−0.321 + 1.70i)3-s + (−1.99 − 0.0776i)4-s + (−1.52 + 1.52i)5-s + (2.39 + 0.501i)6-s − 0.618i·7-s + (−0.164 + 2.82i)8-s + (−2.79 − 1.09i)9-s + (2.11 + 2.20i)10-s + (0.333 − 0.333i)11-s + (0.775 − 3.37i)12-s + (−3.67 − 3.67i)13-s + (−0.874 − 0.0169i)14-s + (−2.10 − 3.08i)15-s + (3.98 + 0.310i)16-s + 3.31i·17-s + ⋯ |
L(s) = 1 | + (0.0194 − 0.999i)2-s + (−0.185 + 0.982i)3-s + (−0.999 − 0.0388i)4-s + (−0.682 + 0.682i)5-s + (0.978 + 0.204i)6-s − 0.233i·7-s + (−0.0582 + 0.998i)8-s + (−0.930 − 0.365i)9-s + (0.669 + 0.695i)10-s + (0.100 − 0.100i)11-s + (0.223 − 0.974i)12-s + (−1.01 − 1.01i)13-s + (−0.233 − 0.00453i)14-s + (−0.543 − 0.797i)15-s + (0.996 + 0.0775i)16-s + 0.802i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.347424 - 0.522464i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.347424 - 0.522464i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0274 + 1.41i)T \) |
| 3 | \( 1 + (0.321 - 1.70i)T \) |
| 19 | \( 1 + (-4.21 + 1.12i)T \) |
good | 5 | \( 1 + (1.52 - 1.52i)T - 5iT^{2} \) |
| 7 | \( 1 + 0.618iT - 7T^{2} \) |
| 11 | \( 1 + (-0.333 + 0.333i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.67 + 3.67i)T + 13iT^{2} \) |
| 17 | \( 1 - 3.31iT - 17T^{2} \) |
| 23 | \( 1 + 2.68T + 23T^{2} \) |
| 29 | \( 1 + (-7.04 + 7.04i)T - 29iT^{2} \) |
| 31 | \( 1 + 6.61iT - 31T^{2} \) |
| 37 | \( 1 + (-2.95 + 2.95i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.83iT - 41T^{2} \) |
| 43 | \( 1 + (4.64 + 4.64i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.73iT - 47T^{2} \) |
| 53 | \( 1 + (6.73 + 6.73i)T + 53iT^{2} \) |
| 59 | \( 1 + (8.83 + 8.83i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.962 - 0.962i)T - 61iT^{2} \) |
| 67 | \( 1 + (-0.863 - 0.863i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.51iT - 71T^{2} \) |
| 73 | \( 1 + 0.167iT - 73T^{2} \) |
| 79 | \( 1 - 8.48iT - 79T^{2} \) |
| 83 | \( 1 + (2.74 + 2.74i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.92iT - 89T^{2} \) |
| 97 | \( 1 - 5.42iT - 97T^{2} \) |
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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
−10.03254461116338483489090118040, −9.416420691225087959210124670437, −8.228683645473783335214280817827, −7.61713908994920960595315993673, −6.09682480749327213550398981494, −5.13197977834109965205050691742, −4.18668835818731977989693081170, −3.42560083559447879384523644549, −2.55582704205119516455915442985, −0.34788553868949039926649016256,
1.21071743729630697930672814666, 3.01717633355282361171816559410, 4.58598242332734195957342300752, 5.08408304590077217251624758960, 6.25850057819676499172693160002, 7.06550676683198304603201802466, 7.64723525552053088034459997615, 8.501156925194978516734080943368, 9.128879203801848036485868451202, 10.10290146933226831790931027890