L(s) = 1 | − i·3-s + 0.913·7-s − 9-s + 3.58i·11-s + 0.913i·13-s − 3.58·17-s − 4i·19-s − 0.913i·21-s + i·27-s − 7.84i·29-s − 5.29·31-s + 3.58·33-s + 7.84i·37-s + 0.913·39-s + 6·41-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.345·7-s − 0.333·9-s + 1.08i·11-s + 0.253i·13-s − 0.868·17-s − 0.917i·19-s − 0.199i·21-s + 0.192i·27-s − 1.45i·29-s − 0.950·31-s + 0.623·33-s + 1.28i·37-s + 0.146·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.08907401918\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08907401918\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.913T + 7T^{2} \) |
| 11 | \( 1 - 3.58iT - 11T^{2} \) |
| 13 | \( 1 - 0.913iT - 13T^{2} \) |
| 17 | \( 1 + 3.58T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 7.84iT - 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 - 7.84iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 7.16iT - 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 2.55iT - 53T^{2} \) |
| 59 | \( 1 + 7.58iT - 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 + 15.1iT - 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 + 5.29T + 79T^{2} \) |
| 83 | \( 1 + 11.1iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 7.16T + 97T^{2} \) |
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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
−7.80139375715080440205517433486, −7.18881925613094673544751113733, −6.53886469729366095965138520938, −5.85221279588221273981591122415, −4.66176596420225870836021370133, −4.47645472993253294559386392331, −3.10938007842150561314790759997, −2.23071372478319383705808512982, −1.48365270231132639989179427389, −0.02310506571313973112157647306,
1.41441182544957268270323360771, 2.55139139634571851209775866199, 3.51852584545500557448710705895, 4.06516028452643451827406553398, 5.12081925939526325950804822246, 5.61682399931325685511583586261, 6.40370932208973629388812172425, 7.29835550771496674022746686121, 8.054809444185196734732943283244, 8.805970017615173641023162281946