L(s) = 1 | + (−0.587 − 0.809i)2-s + (0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.309 − 0.951i)6-s + 2.61i·7-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + (−2.92 + 2.12i)11-s + (−0.587 + 0.809i)12-s + (3.80 − 5.23i)13-s + (2.11 − 1.53i)14-s + (−0.809 − 0.587i)16-s + (−1.17 + 0.381i)17-s − 0.999i·18-s + (1.76 + 5.42i)19-s + ⋯ |
L(s) = 1 | + (−0.415 − 0.572i)2-s + (0.549 + 0.178i)3-s + (−0.154 + 0.475i)4-s + (−0.126 − 0.388i)6-s + 0.989i·7-s + (0.336 − 0.109i)8-s + (0.269 + 0.195i)9-s + (−0.882 + 0.641i)11-s + (−0.169 + 0.233i)12-s + (1.05 − 1.45i)13-s + (0.566 − 0.411i)14-s + (−0.202 − 0.146i)16-s + (−0.285 + 0.0926i)17-s − 0.235i·18-s + (0.404 + 1.24i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28952 + 0.455600i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28952 + 0.455600i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 + 0.809i)T \) |
| 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2.61iT - 7T^{2} \) |
| 11 | \( 1 + (2.92 - 2.12i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.80 + 5.23i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.17 - 0.381i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.76 - 5.42i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.62 - 3.61i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.61 - 8.05i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.04 - 6.29i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.70 + 6.47i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (4.61 + 3.35i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 7.70iT - 43T^{2} \) |
| 47 | \( 1 + (-1.62 - 0.527i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.98 - 0.645i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (2.92 + 2.12i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.23 - 1.62i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.45 + 0.472i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.70 + 5.25i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.07 - 2.85i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.73 + 5.34i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.04 + 0.663i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-2.85 + 2.07i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (3.21 + 1.04i)T + (78.4 + 57.0i)T^{2} \) |
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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
−10.46912308168078689141340423685, −9.586010699521874938037540844201, −8.766693269175112654337077410520, −8.114295384210941482123988474012, −7.34221832183350182157068938335, −5.83110468748523197277077009902, −5.05856299813632001207700502815, −3.55708058928482161505752480015, −2.84116539634868710428113919951, −1.57332820834218130369192555829,
0.804793743482930416750146955707, 2.43001003875025848185115134750, 3.87296611964674414446550338785, 4.76139326360064988823292180865, 6.16632922058012125467512715446, 6.87006364271648120430524555328, 7.70869110173241888762364487214, 8.505744936567859200396282255749, 9.225579290241313725287335607648, 10.09908914001680626683574680923