L(s) = 1 | + (−0.918 − 1.59i)5-s + (0.361 − 2.62i)7-s + (1.54 − 2.68i)11-s + (2.40 − 4.16i)13-s + (−1.87 − 3.24i)17-s + (2.71 − 4.70i)19-s + (3.97 + 6.89i)23-s + (0.813 − 1.40i)25-s + (0.325 + 0.563i)29-s − 1.03·31-s + (−4.50 + 1.83i)35-s + (0.873 − 1.51i)37-s + (−2.52 + 4.36i)41-s + (6.09 + 10.5i)43-s − 4.61·47-s + ⋯ |
L(s) = 1 | + (−0.410 − 0.711i)5-s + (0.136 − 0.990i)7-s + (0.466 − 0.808i)11-s + (0.666 − 1.15i)13-s + (−0.453 − 0.786i)17-s + (0.622 − 1.07i)19-s + (0.829 + 1.43i)23-s + (0.162 − 0.281i)25-s + (0.0604 + 0.104i)29-s − 0.186·31-s + (−0.760 + 0.309i)35-s + (0.143 − 0.248i)37-s + (−0.393 + 0.682i)41-s + (0.929 + 1.61i)43-s − 0.672·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.658107617\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.658107617\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.361 + 2.62i)T \) |
good | 5 | \( 1 + (0.918 + 1.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.54 + 2.68i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.40 + 4.16i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.87 + 3.24i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.71 + 4.70i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.97 - 6.89i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.325 - 0.563i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.03T + 31T^{2} \) |
| 37 | \( 1 + (-0.873 + 1.51i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.52 - 4.36i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.09 - 10.5i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 4.61T + 47T^{2} \) |
| 53 | \( 1 + (4.55 + 7.88i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 5.79T + 59T^{2} \) |
| 61 | \( 1 + 4.81T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 5.00T + 71T^{2} \) |
| 73 | \( 1 + (1.81 + 3.14i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + (-3.83 - 6.63i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.76 + 9.99i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.04 + 1.80i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.324495950655441882705164408167, −7.80270299467403430103537890267, −7.02550827272633936057511470199, −6.20181660054651783637237949242, −5.16566915806583977964924741908, −4.65419143032891493764001280176, −3.57906064367333087731130872586, −3.00860135548033738614656685544, −1.22710298140789278767648783937, −0.59220694381835544696752582918,
1.55785306238091510382877546235, 2.39446903227212145036331763432, 3.49880398765636915222433200030, 4.22407869646262440872269797122, 5.14304487987302305784591664037, 6.19100424042304171843527919030, 6.65584563077476186695697857980, 7.43895555732994318601948568387, 8.349654753453073688017074340231, 8.985914104148701472371834947526