L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s − 0.999·6-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (0.866 − 0.499i)12-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)21-s + (−0.866 + 0.5i)23-s + (−0.499 + 0.866i)24-s − i·27-s + (0.866 − 0.499i)28-s + 29-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s − 0.999·6-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (0.866 − 0.499i)12-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)21-s + (−0.866 + 0.5i)23-s + (−0.499 + 0.866i)24-s − i·27-s + (0.866 − 0.499i)28-s + 29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8943754651\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8943754651\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 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
−10.44411470423099248587524841573, −9.752865560492493156384775781492, −8.914089392972397550639305817495, −8.352042605118239232347399964573, −7.68543330582300518882892333904, −6.49499742689639221533950118616, −5.51095066513700780785236470283, −4.44854473068624014863079163004, −2.99190799930552000473215521646, −1.78730139688979551824940677567,
1.49864871645564382853107396410, 2.48801758194578892004379944530, 3.63245159183965746595912875875, 4.86003982870407861603028904737, 6.51741880116450562745848877659, 7.39854027386070060764693101030, 8.259997633635661931757860217038, 8.454651196230358846672826174242, 9.676356424331704705766518995985, 10.43627920640896071227209184846