| L(s) = 1 | + (0.0465 − 0.00612i)2-s + (0.442 + 0.896i)3-s + (−1.92 + 0.517i)4-s + (3.21 − 2.81i)5-s + (0.0260 + 0.0390i)6-s + (−1.10 + 2.40i)7-s + (−0.173 + 0.0718i)8-s + (−0.608 + 0.793i)9-s + (0.132 − 0.150i)10-s + (4.91 − 0.321i)11-s + (−1.31 − 1.50i)12-s + (1.03 + 1.03i)13-s + (−0.0369 + 0.118i)14-s + (3.95 + 1.63i)15-s + (3.45 − 1.99i)16-s + (−0.291 − 4.11i)17-s + ⋯ |
| L(s) = 1 | + (0.0329 − 0.00433i)2-s + (0.255 + 0.517i)3-s + (−0.964 + 0.258i)4-s + (1.43 − 1.26i)5-s + (0.0106 + 0.0159i)6-s + (−0.419 + 0.907i)7-s + (−0.0612 + 0.0253i)8-s + (−0.202 + 0.264i)9-s + (0.0418 − 0.0477i)10-s + (1.48 − 0.0970i)11-s + (−0.380 − 0.433i)12-s + (0.286 + 0.286i)13-s + (−0.00987 + 0.0316i)14-s + (1.02 + 0.422i)15-s + (0.863 − 0.498i)16-s + (−0.0707 − 0.997i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.51614 + 0.287644i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51614 + 0.287644i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.442 - 0.896i)T \) |
| 7 | \( 1 + (1.10 - 2.40i)T \) |
| 17 | \( 1 + (0.291 + 4.11i)T \) |
| good | 2 | \( 1 + (-0.0465 + 0.00612i)T + (1.93 - 0.517i)T^{2} \) |
| 5 | \( 1 + (-3.21 + 2.81i)T + (0.652 - 4.95i)T^{2} \) |
| 11 | \( 1 + (-4.91 + 0.321i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (-1.03 - 1.03i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.850 - 6.46i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-1.46 - 0.722i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (-0.861 - 4.33i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (1.49 - 0.739i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (0.176 - 2.69i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (1.05 - 5.29i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (1.09 + 2.63i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-1.79 + 6.69i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (4.64 + 6.05i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (12.8 + 1.69i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (3.52 + 10.3i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-4.96 - 2.86i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.57 + 4.39i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (6.39 + 2.16i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (2.81 - 5.70i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (-1.84 + 4.45i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (2.45 + 0.656i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-1.27 + 0.253i)T + (89.6 - 37.1i)T^{2} \) |
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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
−11.83065574048870106429498934239, −10.06807263912755733657342511492, −9.366053468012072986936720136538, −9.068938125406557858709579307973, −8.319750379671872721621462800520, −6.36169830863561470861028829980, −5.45386881207422557746163050153, −4.68068668212970016460031976930, −3.39130576611293901412227652640, −1.56301002683714415550616966153,
1.39076942344610951497298514844, 3.03877285647011284796147085214, 4.22103742430789818296791925851, 5.90395044287763220959329154202, 6.49510360292830114753195850455, 7.38467676382674611624563821498, 8.958504266634949595965028278951, 9.499276931267736371617148870948, 10.39354410250988693994674485676, 11.11125633969570651322327780579
