L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.923 + 0.382i)3-s + 1.00i·4-s + (0.382 − 0.923i)5-s + (−0.923 − 0.382i)6-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + (−0.382 − 0.923i)12-s + (1.30 + 1.30i)13-s + 1.00·14-s + i·15-s − 1.00·16-s + 18-s − 0.765i·19-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.923 + 0.382i)3-s + 1.00i·4-s + (0.382 − 0.923i)5-s + (−0.923 − 0.382i)6-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + (−0.382 − 0.923i)12-s + (1.30 + 1.30i)13-s + 1.00·14-s + i·15-s − 1.00·16-s + 18-s − 0.765i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.221486002\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.221486002\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.923 - 0.382i)T \) |
| 5 | \( 1 + (-0.382 + 0.923i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + 0.765iT - T^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - 0.765T + T^{2} \) |
| 61 | \( 1 + 1.84T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.79193730136006397299967453916, −9.547089552781756117844614625107, −8.828681562710880789735110086842, −7.86526385535305695713282273769, −6.82729055868270970397192204499, −6.06115572596507711562379276359, −5.22509151741996889788123356725, −4.39578220433916340209993048637, −3.84516639899013470651620436230, −1.58331240300568775852231981741,
1.51281901516547494797489065451, 2.61217345257461381476611424368, 3.85114181815973313376553693970, 5.09154197067881635994706699895, 5.99977438097242549376257624509, 6.19552889804234249517259592129, 7.57956017569464895337859526818, 8.575857172733136569761047441610, 9.987319846111477838669689583939, 10.55029568519144654848499301715