L(s) = 1 | + 1.73i·2-s − i·3-s − 1.99·4-s + (−0.866 + 0.5i)5-s + 1.73·6-s − i·7-s − 1.73i·8-s − 9-s + (−0.866 − 1.49i)10-s + 11-s + 1.99i·12-s − i·13-s + 1.73·14-s + (0.5 + 0.866i)15-s + 0.999·16-s + ⋯ |
L(s) = 1 | + 1.73i·2-s − i·3-s − 1.99·4-s + (−0.866 + 0.5i)5-s + 1.73·6-s − i·7-s − 1.73i·8-s − 9-s + (−0.866 − 1.49i)10-s + 11-s + 1.99i·12-s − i·13-s + 1.73·14-s + (0.5 + 0.866i)15-s + 0.999·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7918141561\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7918141561\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 1.73iT - T^{2} \) |
| 13 | \( 1 + iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.73T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.73iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 1.73T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - 1.73iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−9.785116924148163748280683702206, −8.730824394824396448855804245985, −8.007850691369469603584507515739, −7.32840405295998452995668184276, −7.06336513544506538569940177907, −6.17744550234649468148045970213, −5.30049247066631226973566099998, −4.13029624276858724731007393205, −3.16938837403041654713167136728, −0.838687905402583285951691983613,
1.41135047742484393680961276344, 2.91894950487452125863665935181, 3.54365848832387636983011283052, 4.55061992189884802013169431075, 4.98959595128720236955227606167, 6.35547569838516215597372815270, 7.961696125736265884365980330176, 8.916760059974209695243110926343, 9.248172359827823154215752248994, 9.855812894061122638982417420075