L(s) = 1 | + i·3-s + i·7-s − 9-s + 6·11-s + 5i·13-s + 6i·17-s − 5·19-s − 21-s − 6i·23-s − i·27-s + 6·29-s − 31-s + 6i·33-s − 2i·37-s − 5·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.377i·7-s − 0.333·9-s + 1.80·11-s + 1.38i·13-s + 1.45i·17-s − 1.14·19-s − 0.218·21-s − 1.25i·23-s − 0.192i·27-s + 1.11·29-s − 0.179·31-s + 1.04i·33-s − 0.328i·37-s − 0.800·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12912 + 0.697839i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12912 + 0.697839i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + 11iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−11.89775426919520103720214312700, −10.97390440400319704438721262985, −10.02826616092039816278803189986, −8.908658711272978093623584247637, −8.557247793109954747773763479036, −6.71286173103656458170818552512, −6.18398257820960945511517870165, −4.50818170756140184808639903899, −3.82815000832242875890834551587, −1.94481564146603068110691641447,
1.12681038010934193104071281744, 2.96711029637426858672282506267, 4.31922017549397217704922497208, 5.74533468177333226841672193430, 6.76323999610163864840394137867, 7.60226271306038584194672676009, 8.724756441585117969723478982766, 9.623017114521372743848486422670, 10.75249607353648839694489368388, 11.71306478234434015482116738004