L(s) = 1 | + (0.712 + 1.23i)5-s + (2.36 − 1.19i)7-s + (2.46 − 4.27i)11-s + (−1.37 + 2.38i)13-s + (−0.559 − 0.969i)17-s + (2.00 − 3.47i)19-s + (−2.71 − 4.70i)23-s + (1.48 − 2.57i)25-s + (−3.40 − 5.89i)29-s − 2.50·31-s + (3.15 + 2.06i)35-s + (0.709 − 1.22i)37-s + (−0.124 + 0.215i)41-s + (0.498 + 0.863i)43-s − 9.47·47-s + ⋯ |
L(s) = 1 | + (0.318 + 0.551i)5-s + (0.892 − 0.451i)7-s + (0.743 − 1.28i)11-s + (−0.381 + 0.661i)13-s + (−0.135 − 0.235i)17-s + (0.460 − 0.797i)19-s + (−0.566 − 0.981i)23-s + (0.296 − 0.514i)25-s + (−0.632 − 1.09i)29-s − 0.450·31-s + (0.533 + 0.348i)35-s + (0.116 − 0.202i)37-s + (−0.0194 + 0.0336i)41-s + (0.0759 + 0.131i)43-s − 1.38·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.006528217\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.006528217\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.36 + 1.19i)T \) |
good | 5 | \( 1 + (-0.712 - 1.23i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.46 + 4.27i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.37 - 2.38i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.559 + 0.969i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.00 + 3.47i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.71 + 4.70i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.40 + 5.89i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 + (-0.709 + 1.22i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.124 - 0.215i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.498 - 0.863i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 9.47T + 47T^{2} \) |
| 53 | \( 1 + (-0.410 - 0.710i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 6.58T + 59T^{2} \) |
| 61 | \( 1 - 0.0752T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 0.0804T + 71T^{2} \) |
| 73 | \( 1 + (-5.34 - 9.25i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 1.84T + 79T^{2} \) |
| 83 | \( 1 + (7.23 + 12.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.76 - 11.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.70 - 4.67i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.527696950597654041334164896898, −7.903619172270740214318486678561, −6.94821978216258596193968874074, −6.44355506231749254201673363308, −5.57241247496281633646740836557, −4.61424435522477046089365920018, −3.91107742814149410555035423141, −2.83947015808897309819118902984, −1.90574999421930746477905986473, −0.63135890695479856478543374021,
1.44859730147662817735110059670, 1.91178267895621716900892122160, 3.31232852500837920207624014562, 4.27757187397875415319627991160, 5.18032458998077463183256893438, 5.51424243533291368275724228644, 6.64434591762537254911773110939, 7.51628550386104017385116397759, 8.034821493538708699685243456234, 8.961754974511951920924445402852