| L(s) = 1 | + 1.83i·3-s + (−1.86 − 1.23i)5-s − i·7-s − 0.359·9-s − 4.40·11-s − 3.20i·13-s + (2.26 − 3.41i)15-s + 1.14i·17-s − 1.72·19-s + 1.83·21-s + 3.25i·23-s + (1.94 + 4.60i)25-s + 4.83i·27-s + 4.18·29-s + 1.36·31-s + ⋯ |
| L(s) = 1 | + 1.05i·3-s + (−0.833 − 0.553i)5-s − 0.377i·7-s − 0.119·9-s − 1.32·11-s − 0.889i·13-s + (0.585 − 0.881i)15-s + 0.278i·17-s − 0.395·19-s + 0.399·21-s + 0.678i·23-s + (0.388 + 0.921i)25-s + 0.931i·27-s + 0.777·29-s + 0.245·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.232612232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.232612232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.86 + 1.23i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 - 1.83iT - 3T^{2} \) |
| 11 | \( 1 + 4.40T + 11T^{2} \) |
| 13 | \( 1 + 3.20iT - 13T^{2} \) |
| 17 | \( 1 - 1.14iT - 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 23 | \( 1 - 3.25iT - 23T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 31 | \( 1 - 1.36T + 31T^{2} \) |
| 37 | \( 1 + 4.19iT - 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 2.64iT - 43T^{2} \) |
| 47 | \( 1 + 0.106iT - 47T^{2} \) |
| 53 | \( 1 - 7.86iT - 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 1.28iT - 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 11.8iT - 73T^{2} \) |
| 79 | \( 1 + 5.48T + 79T^{2} \) |
| 83 | \( 1 - 17.1iT - 83T^{2} \) |
| 89 | \( 1 + 4.31T + 89T^{2} \) |
| 97 | \( 1 - 7.65iT - 97T^{2} \) |
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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
−9.255837974710400587739848755205, −8.262033903078137877765778388192, −7.85261305215361423055461326413, −7.01561226203849407650681592537, −5.66283617564840119086694295988, −5.07479543522892047842681080570, −4.27546620346987276136925132014, −3.64046267008846383654514642880, −2.61646903385186984852304792949, −0.828240065267580172211029408471,
0.60856026000982959673843974147, 2.17961058780492638334941723488, 2.75428563485417695094831362917, 4.04099845887598017787277263819, 4.88248995517174335752022454715, 6.00347047281250020422774169761, 6.83873425804892823680048710130, 7.22727287603223300843067560417, 8.195063293385082552195010199789, 8.453351169077375777083441468520
