L(s) = 1 | + i·5-s − 9-s + 2i·13-s − 25-s + 2i·37-s + 2·41-s − i·45-s − 49-s − 2i·53-s − 2·65-s + 81-s + 2·89-s − 2i·117-s + ⋯ |
L(s) = 1 | + i·5-s − 9-s + 2i·13-s − 25-s + 2i·37-s + 2·41-s − i·45-s − 49-s − 2i·53-s − 2·65-s + 81-s + 2·89-s − 2i·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8890633097\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8890633097\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
good | 3 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 2iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 2iT - T^{2} \) |
| 41 | \( 1 - 2T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 2iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 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
−9.994793330505091035249915327841, −9.316453262918297456476663793718, −8.480920221106173447274895925969, −7.57727930770357627521406680542, −6.59957908161261979085605633843, −6.22696551965542421121709836129, −4.97860869551230558678845950276, −3.94044137499837100849588764277, −2.93826207171072861338336841391, −1.94926018263079764257266318814,
0.75773930440641334800691844760, 2.45287994393897525826934181591, 3.48531571932710923668761114259, 4.64458366134901013714509863897, 5.63658752206409482313247284201, 5.92421089632189173095394856323, 7.54289146383642021304464295970, 8.026880209641628523569417145420, 8.869293515343342907580846639575, 9.482603458082583483086456790282