L(s) = 1 | + 4.24i·7-s + 4i·11-s − 6i·13-s − 4.24i·17-s − 2.82·19-s − 6·23-s − 5·25-s + 8.48·29-s + 4.24i·31-s + 6i·37-s − 1.41i·41-s + 2.82·43-s − 6·47-s − 10.9·49-s − 8.48·53-s + ⋯ |
L(s) = 1 | + 1.60i·7-s + 1.20i·11-s − 1.66i·13-s − 1.02i·17-s − 0.648·19-s − 1.25·23-s − 25-s + 1.57·29-s + 0.762i·31-s + 0.986i·37-s − 0.220i·41-s + 0.431·43-s − 0.875·47-s − 1.57·49-s − 1.16·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 4.24iT - 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 4.24iT - 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 8.48T + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 4.24iT - 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + 12.7iT - 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−8.125974237651771883983128412779, −7.42554559898036222152035001544, −6.41647818701946783047109908702, −5.84514082700969695599052517028, −5.07190287280669263140621860441, −4.49540233511724341030091405801, −3.15058284072668218557616171973, −2.58307213889633054653658824447, −1.68399398458318785359351424723, 0,
1.24461563731805755253719800654, 2.19529151494564890279757568396, 3.56997456721862409932490842855, 4.06987094522080049885040666844, 4.60592248666457152763384302168, 6.02433411631433385037643080235, 6.33061324666712569202371284553, 7.13288330128148443182341915523, 8.002183009253166077976876169962