L(s) = 1 | + (−0.831 − 0.831i)2-s + (0.763 + 1.84i)3-s + 0.381i·4-s + (0.896 − 2.16i)6-s + (0.431 + 0.178i)7-s + (−0.513 + 0.513i)8-s + (−2.10 + 2.10i)9-s + (−0.703 + 0.291i)12-s + (−0.210 − 0.507i)14-s + 1.23·16-s + (−0.309 + 0.951i)17-s + 3.49·18-s + 0.930i·21-s + (−1.33 − 0.554i)24-s + (0.707 − 0.707i)25-s + ⋯ |
L(s) = 1 | + (−0.831 − 0.831i)2-s + (0.763 + 1.84i)3-s + 0.381i·4-s + (0.896 − 2.16i)6-s + (0.431 + 0.178i)7-s + (−0.513 + 0.513i)8-s + (−2.10 + 2.10i)9-s + (−0.703 + 0.291i)12-s + (−0.210 − 0.507i)14-s + 1.23·16-s + (−0.309 + 0.951i)17-s + 3.49·18-s + 0.930i·21-s + (−1.33 − 0.554i)24-s + (0.707 − 0.707i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7563757561\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7563757561\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (0.309 - 0.951i)T \) |
| 47 | \( 1 - iT \) |
good | 2 | \( 1 + (0.831 + 0.831i)T + iT^{2} \) |
| 3 | \( 1 + (-0.763 - 1.84i)T + (-0.707 + 0.707i)T^{2} \) |
| 5 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.431 - 0.178i)T + (0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (0.0600 + 0.144i)T + (-0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1.39 - 1.39i)T + iT^{2} \) |
| 59 | \( 1 + (-1.14 + 1.14i)T - iT^{2} \) |
| 61 | \( 1 + (-0.965 - 0.399i)T + (0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.581 + 1.40i)T + (-0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (0.581 - 1.40i)T + (-0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 + (1 + i)T + iT^{2} \) |
| 89 | \( 1 + 1.17iT - T^{2} \) |
| 97 | \( 1 + (-1.57 + 0.652i)T + (0.707 - 0.707i)T^{2} \) |
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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
−10.31096306975160726622657191838, −10.03330669101625114048153909436, −8.911480319403287142674071878136, −8.712281745340990767208863974441, −7.84853695305949174057369749948, −5.96144313812804438660231382226, −5.01365341296172298303341089309, −4.08743749776862720098052670117, −3.03746057365108343043860821507, −2.09975073747178200638069797952,
0.983077308087567981076856614977, 2.40324343431032421921264042041, 3.51776092280825576346281535218, 5.44894200584658576524108993443, 6.53717174380604937073566680594, 7.13408209796798926846644936894, 7.60983983031758183388215094648, 8.559719564237631442727363349293, 8.849007342980339223311596788531, 9.894489761527655202307079627021