L(s) = 1 | + (1.11 + 1.92i)5-s + (−1.78 − 1.95i)7-s + (−1.51 − 0.876i)11-s + (3.37 − 1.94i)13-s − 1.78·17-s − 2.73i·19-s + (1.64 − 0.947i)23-s + (0.0196 − 0.0339i)25-s + (7.48 + 4.32i)29-s + (−7.18 + 4.14i)31-s + (1.78 − 5.61i)35-s − 9.91·37-s + (−3.42 − 5.93i)41-s + (4.22 − 7.32i)43-s + (−1.47 + 2.54i)47-s + ⋯ |
L(s) = 1 | + (0.498 + 0.862i)5-s + (−0.673 − 0.738i)7-s + (−0.457 − 0.264i)11-s + (0.935 − 0.540i)13-s − 0.433·17-s − 0.628i·19-s + (0.342 − 0.197i)23-s + (0.00392 − 0.00678i)25-s + (1.39 + 0.802i)29-s + (−1.29 + 0.745i)31-s + (0.301 − 0.949i)35-s − 1.63·37-s + (−0.535 − 0.927i)41-s + (0.644 − 1.11i)43-s + (−0.214 + 0.371i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0755 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0755 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.310420747\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.310420747\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.78 + 1.95i)T \) |
good | 5 | \( 1 + (-1.11 - 1.92i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.51 + 0.876i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.37 + 1.94i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.78T + 17T^{2} \) |
| 19 | \( 1 + 2.73iT - 19T^{2} \) |
| 23 | \( 1 + (-1.64 + 0.947i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.48 - 4.32i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (7.18 - 4.14i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 9.91T + 37T^{2} \) |
| 41 | \( 1 + (3.42 + 5.93i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.22 + 7.32i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.47 - 2.54i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 5.45iT - 53T^{2} \) |
| 59 | \( 1 + (-0.449 - 0.778i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (10.3 + 5.98i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.38 + 9.32i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14.2iT - 71T^{2} \) |
| 73 | \( 1 - 4.71iT - 73T^{2} \) |
| 79 | \( 1 + (-4.70 + 8.15i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.326 + 0.565i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 9.70T + 89T^{2} \) |
| 97 | \( 1 + (-0.0294 - 0.0169i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.703017396238953149662227891139, −7.65814074004242203079364871064, −6.83088278084310211605339432926, −6.51760084921996640901990257852, −5.56720973706895160350066636690, −4.70105869306826523046511529150, −3.41439690486347638428455145040, −3.12922315998917384013932670525, −1.86750488733206491343870056679, −0.41657389995443660920423668082,
1.26444175072654481982041281223, 2.23517525132952613864515571686, 3.27393147878357634071753208148, 4.28551464895448491423534057561, 5.14340889405746739175353428616, 5.87879281755327244368221841134, 6.43999974245807205564564124404, 7.41038994307098504946910819052, 8.437384816028051458568729721068, 8.861095783076613755459976326101