| L(s) = 1 | − 1.55·5-s + 1.45i·7-s + 5.49i·11-s + 1.16i·13-s − 2.73i·17-s + 7.56·19-s + 23-s − 2.57·25-s + 6.44·29-s − 5.80i·31-s − 2.27i·35-s − 10.7i·37-s − 2.52i·41-s + 10.8·43-s − 5.81·47-s + ⋯ |
| L(s) = 1 | − 0.696·5-s + 0.551i·7-s + 1.65i·11-s + 0.323i·13-s − 0.663i·17-s + 1.73·19-s + 0.208·23-s − 0.514·25-s + 1.19·29-s − 1.04i·31-s − 0.384i·35-s − 1.76i·37-s − 0.393i·41-s + 1.64·43-s − 0.848·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.716562126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.716562126\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 1.55T + 5T^{2} \) |
| 7 | \( 1 - 1.45iT - 7T^{2} \) |
| 11 | \( 1 - 5.49iT - 11T^{2} \) |
| 13 | \( 1 - 1.16iT - 13T^{2} \) |
| 17 | \( 1 + 2.73iT - 17T^{2} \) |
| 19 | \( 1 - 7.56T + 19T^{2} \) |
| 29 | \( 1 - 6.44T + 29T^{2} \) |
| 31 | \( 1 + 5.80iT - 31T^{2} \) |
| 37 | \( 1 + 10.7iT - 37T^{2} \) |
| 41 | \( 1 + 2.52iT - 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 5.81T + 47T^{2} \) |
| 53 | \( 1 - 6.30T + 53T^{2} \) |
| 59 | \( 1 - 2.95iT - 59T^{2} \) |
| 61 | \( 1 - 1.73iT - 61T^{2} \) |
| 67 | \( 1 + 8.18T + 67T^{2} \) |
| 71 | \( 1 + 2.64T + 71T^{2} \) |
| 73 | \( 1 + 2.61T + 73T^{2} \) |
| 79 | \( 1 - 14.4iT - 79T^{2} \) |
| 83 | \( 1 - 17.7iT - 83T^{2} \) |
| 89 | \( 1 - 2.77iT - 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.932596987164150583386974781465, −7.30870954297359281892411456671, −7.07065954179675789328624800252, −5.87875576167249696177236345874, −5.27276665874299248822140300671, −4.43458208434614179672254739119, −3.90237639839185637973841617875, −2.77756760060252407064973758898, −2.12864176465938123744926558165, −0.864663397488098881940412808158,
0.59876614944239314883798891326, 1.34564222946888461295752444083, 3.06119049677718100071695935730, 3.24458808213398140614899463577, 4.18970115620674152305580607096, 5.00063685960250580202311672820, 5.81789223802303306389562466506, 6.42081593129051912047415065591, 7.34297491622557478976530192144, 7.86450343372938450435701422329
