L(s) = 1 | + (0.923 + 0.382i)2-s + (0.707 + 0.707i)4-s + (−1.08 + 1.63i)5-s + (0.382 + 0.923i)8-s + (0.923 − 0.382i)9-s + (−1.63 + 1.08i)10-s + (1 + i)13-s + i·16-s + (0.382 − 0.923i)17-s + 18-s + (−1.92 + 0.382i)20-s + (−1.08 − 2.63i)25-s + (0.541 + 1.30i)26-s + (−1.08 + 1.63i)29-s + (−0.382 + 0.923i)32-s + ⋯ |
L(s) = 1 | + (0.923 + 0.382i)2-s + (0.707 + 0.707i)4-s + (−1.08 + 1.63i)5-s + (0.382 + 0.923i)8-s + (0.923 − 0.382i)9-s + (−1.63 + 1.08i)10-s + (1 + i)13-s + i·16-s + (0.382 − 0.923i)17-s + 18-s + (−1.92 + 0.382i)20-s + (−1.08 − 2.63i)25-s + (0.541 + 1.30i)26-s + (−1.08 + 1.63i)29-s + (−0.382 + 0.923i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.021588738\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.021588738\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.923 - 0.382i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.382 + 0.923i)T \) |
good | 3 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 5 | \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \) |
| 11 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 19 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 29 | \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 37 | \( 1 + (0.0761 + 0.382i)T + (-0.923 + 0.382i)T^{2} \) |
| 41 | \( 1 + (0.617 + 0.923i)T + (-0.382 + 0.923i)T^{2} \) |
| 43 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (-1.38 + 0.923i)T + (0.382 - 0.923i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 73 | \( 1 + (0.216 - 0.324i)T + (-0.382 - 0.923i)T^{2} \) |
| 79 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 + (0.324 + 0.216i)T + (0.382 + 0.923i)T^{2} \) |
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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
−8.873163191442312574555102705512, −7.957344511240910119031020315797, −7.13289820016826414554381244123, −6.97311172102294798257766378507, −6.29807176386833973739071339029, −5.22129955248850961001724175281, −4.13971111097872947451759548363, −3.68409307183499473672401296016, −3.05781310144899554919331116070, −1.86896000179251455647016839950,
0.965159664322735468485494489528, 1.76327183261038844085490232735, 3.33368511260173386938354206174, 4.03756690024741070624029271108, 4.50028188772951555769425019495, 5.39729407465137913512308319893, 5.94357306501586179954108384983, 7.11204902522779192837878863050, 8.024094757843455324818173765056, 8.229858595683774908524804548909