L(s) = 1 | + i·3-s + i·5-s − 1.61·7-s − 9-s − 1.15·11-s + 7.10·13-s − 15-s − 1.79i·17-s − 5.93·19-s − 1.61i·21-s + (−4.40 − 1.89i)23-s − 25-s − i·27-s − 0.0365·29-s + 4.39i·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.447i·5-s − 0.609·7-s − 0.333·9-s − 0.346·11-s + 1.97·13-s − 0.258·15-s − 0.434i·17-s − 1.36·19-s − 0.351i·21-s + (−0.918 − 0.395i)23-s − 0.200·25-s − 0.192i·27-s − 0.00679·29-s + 0.789i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.386596061\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.386596061\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 23 | \( 1 + (4.40 + 1.89i)T \) |
good | 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 13 | \( 1 - 7.10T + 13T^{2} \) |
| 17 | \( 1 + 1.79iT - 17T^{2} \) |
| 19 | \( 1 + 5.93T + 19T^{2} \) |
| 29 | \( 1 + 0.0365T + 29T^{2} \) |
| 31 | \( 1 - 4.39iT - 31T^{2} \) |
| 37 | \( 1 + 10.1iT - 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 - 9.60T + 43T^{2} \) |
| 47 | \( 1 + 9.16iT - 47T^{2} \) |
| 53 | \( 1 + 1.95iT - 53T^{2} \) |
| 59 | \( 1 + 5.29iT - 59T^{2} \) |
| 61 | \( 1 - 9.53iT - 61T^{2} \) |
| 67 | \( 1 - 2.33T + 67T^{2} \) |
| 71 | \( 1 + 11.3iT - 71T^{2} \) |
| 73 | \( 1 + 1.61T + 73T^{2} \) |
| 79 | \( 1 - 5.33T + 79T^{2} \) |
| 83 | \( 1 - 0.722T + 83T^{2} \) |
| 89 | \( 1 - 2.09iT - 89T^{2} \) |
| 97 | \( 1 + 1.93iT - 97T^{2} \) |
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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
−8.258557308168757794301672951866, −7.39652714133598161690702626691, −6.42408204759593232073086205774, −6.11671998804762450915493567622, −5.30496117654228583775571422427, −4.13204587671313374324146393152, −3.79186262729980757578816117301, −2.87437352652308085635460015467, −1.95935012065823213345418196377, −0.43770958148564349140230039713,
0.909025478744700049156887050374, 1.81804114457543391763559866390, 2.83074598331222616019293132557, 3.81299997677443696634781872084, 4.36067683048770041820642940750, 5.58191582236128222026512829703, 6.26856177571455666790412198142, 6.42392643703815654251523025035, 7.69411126460975596614827383374, 8.159589940125641535048185236365