L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.741 + 1.28i)3-s + (−0.499 − 0.866i)4-s + (0.741 + 1.28i)6-s − 0.482·7-s − 0.999·8-s + (0.400 + 0.693i)9-s − 4.43·11-s + 1.48·12-s + (−2.07 − 3.58i)13-s + (−0.241 + 0.418i)14-s + (−0.5 + 0.866i)16-s + (3.94 − 6.84i)17-s + 0.801·18-s + (4.31 + 0.590i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.428 + 0.741i)3-s + (−0.249 − 0.433i)4-s + (0.302 + 0.524i)6-s − 0.182·7-s − 0.353·8-s + (0.133 + 0.231i)9-s − 1.33·11-s + 0.428·12-s + (−0.574 − 0.994i)13-s + (−0.0645 + 0.111i)14-s + (−0.125 + 0.216i)16-s + (0.957 − 1.65i)17-s + 0.188·18-s + (0.990 + 0.135i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.431 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.526528 - 0.835901i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.526528 - 0.835901i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-4.31 - 0.590i)T \) |
good | 3 | \( 1 + (0.741 - 1.28i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 0.482T + 7T^{2} \) |
| 11 | \( 1 + 4.43T + 11T^{2} \) |
| 13 | \( 1 + (2.07 + 3.58i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.94 + 6.84i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (2.82 + 4.90i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.91 + 3.31i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.58T + 31T^{2} \) |
| 37 | \( 1 + 3.50T + 37T^{2} \) |
| 41 | \( 1 + (-4.44 + 7.70i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.318 + 0.551i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.13 - 3.69i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.59 + 2.76i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.812 - 1.40i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.735 + 1.27i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.59 + 11.4i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.41 - 2.44i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.43 + 5.95i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.576 + 0.997i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + (4.37 + 7.58i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.476 + 0.825i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.13119823060410553581722800937, −9.390255592253512280665887408314, −7.977736064698413461091821972167, −7.42230602093263606487128812251, −5.86739761564147567737272054206, −5.16298178067737065872467396393, −4.64246406751294526822753503812, −3.25736262727471465078150319555, −2.48328317857295522794370425816, −0.43635051499449360878986528826,
1.52149927251986706653108394257, 3.05872028856277631821557257919, 4.20219886036276935739782000853, 5.38100616254090652991364278710, 5.99793628805771921440183803103, 6.92057907279595052291285310656, 7.62413809424541398845772412973, 8.268810641194130464101559647698, 9.542383893167021773700990047366, 10.15328892412261477511239221792