L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.133 − 2.23i)5-s + 0.999·6-s + (3.46 − 2i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−1 + 1.99i)10-s + 11-s + (−0.866 − 0.499i)12-s + (2.59 − 1.5i)13-s − 3.99·14-s + (1.23 + 1.86i)15-s + (−0.5 + 0.866i)16-s + (5.19 + 3i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.0599 − 0.998i)5-s + 0.408·6-s + (1.30 − 0.755i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.316 + 0.632i)10-s + 0.301·11-s + (−0.249 − 0.144i)12-s + (0.720 − 0.416i)13-s − 1.06·14-s + (0.318 + 0.481i)15-s + (−0.125 + 0.216i)16-s + (1.26 + 0.727i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.257731578\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.257731578\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.133 + 2.23i)T \) |
| 37 | \( 1 + (-6.06 + 0.5i)T \) |
good | 7 | \( 1 + (-3.46 + 2i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + (-2.59 + 1.5i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-5.19 - 3i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 7iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 41 | \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 + (-1.73 - i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.5 + 9.52i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.66 + 5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.66 + 5i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.5 + 4.33i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−9.786953146904648968094520057249, −8.855100372255063372804950449678, −7.945764068661464065915844084033, −7.71560438804395283896826434345, −6.16583355125133414259685367787, −5.35002193860007193946744069289, −4.33702238561373734486307242603, −3.63639334672869551973952218023, −1.64779610260291178492682258260, −0.901506178404924497312163161725,
1.29723956963573633539351893812, 2.40680611933508535784697927084, 3.80971575295479733257891132704, 5.40468767264128944277481186789, 5.62723625092631929489488712138, 6.94286987516285565539101555076, 7.42849918603980650563846430249, 8.214418755749722636590122841934, 9.219279198444351189172927720850, 9.897590221460543995883513576974