L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.866 + 0.5i)3-s − 1.00i·4-s + (−0.965 + 0.258i)5-s + (−0.965 + 0.258i)6-s + (1.22 + 1.22i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (0.500 − 0.866i)10-s + (0.500 − 0.866i)12-s + (0.707 − 0.707i)13-s − 1.73·14-s + (−0.965 − 0.258i)15-s − 1.00·16-s + (0.366 − 0.366i)17-s + (−0.965 − 0.258i)18-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.866 + 0.5i)3-s − 1.00i·4-s + (−0.965 + 0.258i)5-s + (−0.965 + 0.258i)6-s + (1.22 + 1.22i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (0.500 − 0.866i)10-s + (0.500 − 0.866i)12-s + (0.707 − 0.707i)13-s − 1.73·14-s + (−0.965 − 0.258i)15-s − 1.00·16-s + (0.366 − 0.366i)17-s + (−0.965 − 0.258i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.041904593\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.041904593\) |
\(L(1)\) |
|
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 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.965 - 0.258i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.366 + 0.366i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1.36 - 1.36i)T - iT^{2} \) |
| 47 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 1.93iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.583884255788750230295322362295, −8.691706038346407946632301642160, −8.414755148550003869309455649118, −7.74710403880205883995147631439, −7.03736519889212983785620253071, −5.63054110029849781059180572784, −5.09182736237189084485056870341, −4.03057737409857524594789710649, −2.84591030321220175673357460901, −1.68453519990672647516195847408,
1.10843715516332048337933947853, 1.89499360570882308499042798779, 3.49481123314592059753316284259, 3.90027748880763616123361455862, 4.79745752270184273446937786187, 6.69301881964880724106949155908, 7.47469211398851445243200551057, 7.83481507805389414182958647113, 8.621137196602756296930638681225, 9.095045662079452547547592645149