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
\(L(1)\) |
\(\approx\) |
\(2.651641860\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.651641860\) |
\(L(\frac{3}{2})\) |
|
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
−7.966569311234834337350947737992, −7.42396812502650414188266800179, −6.76106504602574190410466067914, −5.92010795082467997625562519977, −5.47585509438577284367995902857, −4.35968851994020744496984570620, −3.52454054883753927213149895919, −2.80632181808885109159443878246, −1.60988639032514052797460496266, −0.944196826992746260152581432610,
0.996388751017456797157900043408, 1.69409759043281800428811218654, 3.24628293702101758923767121936, 3.59986722356540175725407625713, 4.71543514566262966986361726782, 5.09442950473869721433162876230, 6.04869132072560624694621163625, 6.61578846233436334103537297982, 7.65071964401926719483719183358, 8.508883719356104840392398156469