L(s) = 1 | − i·3-s + (−1.81 + 1.30i)5-s − 3.73i·7-s − 9-s − 5.75·11-s + 4.85i·13-s + (1.30 + 1.81i)15-s + 5.12i·17-s + 8.11·19-s − 3.73·21-s + i·23-s + (1.59 − 4.73i)25-s + i·27-s + 0.516·29-s + 6.47·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (−0.812 + 0.583i)5-s − 1.41i·7-s − 0.333·9-s − 1.73·11-s + 1.34i·13-s + (0.336 + 0.468i)15-s + 1.24i·17-s + 1.86·19-s − 0.815·21-s + 0.208i·23-s + (0.319 − 0.947i)25-s + 0.192i·27-s + 0.0958·29-s + 1.16·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.038208568\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.038208568\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.81 - 1.30i)T \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 + 3.73iT - 7T^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 - 4.85iT - 13T^{2} \) |
| 17 | \( 1 - 5.12iT - 17T^{2} \) |
| 19 | \( 1 - 8.11T + 19T^{2} \) |
| 29 | \( 1 - 0.516T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 - 7.07iT - 37T^{2} \) |
| 41 | \( 1 - 4.42T + 41T^{2} \) |
| 43 | \( 1 + 1.54iT - 43T^{2} \) |
| 47 | \( 1 + 10.0iT - 47T^{2} \) |
| 53 | \( 1 - 2.22iT - 53T^{2} \) |
| 59 | \( 1 - 7.93T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 8.86iT - 67T^{2} \) |
| 71 | \( 1 + 5.93T + 71T^{2} \) |
| 73 | \( 1 - 14.3iT - 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 - 3.73iT - 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 0.693iT - 97T^{2} \) |
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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
−10.10670036496092057138888869810, −8.513303326253803856165549237779, −7.895745224946270611446863016123, −7.20740366773390946643734165531, −6.76904028466109868146181897942, −5.52816529514912817497920090702, −4.42289476487023855686231939583, −3.60087417495698127602602327084, −2.56369689044956989303289260275, −1.04779762186303582733737037476,
0.51749627356836044623511710196, 2.73121315822957519899411472094, 3.08960455658588820552131571631, 4.65926930195185049305817095711, 5.34387541667763616193136058652, 5.68840112201187678074618498455, 7.44003477480298076317892910794, 7.926345014823546773277607891616, 8.677132135858447753603175705240, 9.485562732452676682101980145495