L(s) = 1 | − i·2-s + (0.707 + 0.707i)3-s − 4-s + (−1 − i)5-s + (0.707 − 0.707i)6-s + i·7-s + i·8-s + 1.00i·9-s + (−1 + i)10-s + (0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s + (0.707 + 0.707i)13-s + 14-s − 1.41i·15-s + 16-s + 17-s + ⋯ |
L(s) = 1 | − i·2-s + (0.707 + 0.707i)3-s − 4-s + (−1 − i)5-s + (0.707 − 0.707i)6-s + i·7-s + i·8-s + 1.00i·9-s + (−1 + i)10-s + (0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s + (0.707 + 0.707i)13-s + 14-s − 1.41i·15-s + 16-s + 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.156647003\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.156647003\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 37 | \( 1 + (-0.707 + 0.707i)T \) |
good | 5 | \( 1 + (1 + i)T + iT^{2} \) |
| 7 | \( 1 - iT - T^{2} \) |
| 11 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 13 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 - 1.41iT - T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 59 | \( 1 + (-1 - i)T + iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 89 | \( 1 + iT - T^{2} \) |
| 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
−9.193914986125483523397936238739, −8.686379503229682632802842170519, −8.547385493966478068082054653004, −7.44624229228111263284162194550, −5.79606467570817623557131077184, −5.07184312468271692061769994372, −4.08959093289878904249730327305, −3.68015090966488640776797502712, −2.61656112252936781010508748306, −1.31628601342381893272028984526,
1.02205453255516235267879188464, 2.88409717561937971908460859770, 3.95533944100844828531669059399, 4.15834993489228568097480162240, 5.96626695592450354259384919551, 6.55431276306578556216345848095, 7.32679850238059760365951089402, 7.80049876604908852406552877459, 8.252845200728925715451259147388, 9.360361591165990799322227165982