L(s) = 1 | + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s + (−0.923 − 0.382i)7-s + (−0.923 + 0.382i)8-s + (0.923 − 0.382i)9-s + (0.324 − 1.63i)11-s + (−0.707 + 0.707i)14-s + i·16-s − i·18-s + (−1.38 − 0.923i)22-s + (0.707 + 1.70i)23-s + (−0.382 + 0.923i)25-s + (0.382 + 0.923i)28-s + (0.216 + 1.08i)29-s + (0.923 + 0.382i)32-s + ⋯ |
L(s) = 1 | + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s + (−0.923 − 0.382i)7-s + (−0.923 + 0.382i)8-s + (0.923 − 0.382i)9-s + (0.324 − 1.63i)11-s + (−0.707 + 0.707i)14-s + i·16-s − i·18-s + (−1.38 − 0.923i)22-s + (0.707 + 1.70i)23-s + (−0.382 + 0.923i)25-s + (0.382 + 0.923i)28-s + (0.216 + 1.08i)29-s + (0.923 + 0.382i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9153627997\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9153627997\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.382 + 0.923i)T \) |
| 7 | \( 1 + (0.923 + 0.382i)T \) |
good | 3 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 5 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 11 | \( 1 + (-0.324 + 1.63i)T + (-0.923 - 0.382i)T^{2} \) |
| 13 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 23 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \) |
| 41 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-0.382 + 1.92i)T + (-0.923 - 0.382i)T^{2} \) |
| 59 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 61 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 67 | \( 1 + (1.92 - 0.382i)T + (0.923 - 0.382i)T^{2} \) |
| 71 | \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 83 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 89 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−11.13514997682342806357188859709, −10.22517330402765310952347166300, −9.451934533169859489030986156027, −8.726505306528812413204516739893, −7.21785965465829034292177828359, −6.20107513393605711136734091612, −5.20708169082076706543131504249, −3.69589403950151495664122088322, −3.29279849857578592394624745439, −1.27416153341040169943284816662,
2.51143894717075304928801970058, 4.13307241291556885672749716365, 4.77629470737715626755322504908, 6.18073434917292392613427763235, 6.87535110408835168306397938790, 7.66380835195516936024913954204, 8.827714643004513568855999051045, 9.703128394198623611340275720439, 10.35975306681417962045267137553, 12.11639916572497631634618615686