L(s) = 1 | + i·2-s + 1.73i·3-s − 4-s + (−0.866 + 0.5i)5-s − 1.73·6-s − i·7-s − i·8-s − 1.99·9-s + (−0.5 − 0.866i)10-s − 1.73i·12-s + i·13-s + 14-s + (−0.866 − 1.49i)15-s + 16-s + 1.73i·17-s − 1.99i·18-s + ⋯ |
L(s) = 1 | + i·2-s + 1.73i·3-s − 4-s + (−0.866 + 0.5i)5-s − 1.73·6-s − i·7-s − i·8-s − 1.99·9-s + (−0.5 − 0.866i)10-s − 1.73i·12-s + i·13-s + 14-s + (−0.866 − 1.49i)15-s + 16-s + 1.73i·17-s − 1.99i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6001626969\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6001626969\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 - iT \) |
good | 3 | \( 1 - 1.73iT - T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.73iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.73iT - T^{2} \) |
| 47 | \( 1 - iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - 1.73T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 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
−11.20453362563998891286111706179, −10.50776795271031989637944295138, −9.903573168737151660809090073130, −8.852888155067407494795786188189, −8.139981617081447990603193895780, −7.06208857829881901399829622480, −6.10248182120633860567740431736, −4.78434119702446619747912220440, −4.04602898390995568251932601569, −3.57527686954444699339730162436,
0.77736048504211032404345062386, 2.30363656760984094385664778625, 3.20800200293129387233130267401, 4.94478161083362709636856493087, 5.78625668128891146151644445050, 7.21669653344067031509391657958, 7.999479145416817827579433416723, 8.667419915981627458262782558309, 9.538716072293869519886231500739, 11.09008642757580458692221991613