L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 4.04·5-s + (−0.499 − 0.866i)6-s + (0.346 + 0.599i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (2.02 − 3.50i)10-s + (−2.42 + 4.20i)11-s + 0.999·12-s − 0.692·14-s + (2.02 − 3.50i)15-s + (−0.5 + 0.866i)16-s + (−3.69 − 6.39i)17-s + 0.999·18-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s − 1.81·5-s + (−0.204 − 0.353i)6-s + (0.130 + 0.226i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.640 − 1.10i)10-s + (−0.731 + 1.26i)11-s + 0.288·12-s − 0.184·14-s + (0.522 − 0.905i)15-s + (−0.125 + 0.216i)16-s + (−0.895 − 1.55i)17-s + 0.235·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4056734388\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4056734388\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 4.04T + 5T^{2} \) |
| 7 | \( 1 + (-0.346 - 0.599i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.42 - 4.20i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3.69 + 6.39i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.890 + 1.54i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.55 - 4.42i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.67 + 2.89i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.972T + 31T^{2} \) |
| 37 | \( 1 + (-0.643 + 1.11i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.753 + 1.30i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.15 - 7.20i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 7.20T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + (-0.653 - 1.13i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.198 - 0.343i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.02 + 5.24i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.664 + 1.15i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 7.65T + 73T^{2} \) |
| 79 | \( 1 + 8.33T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + (-1.55 + 2.69i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.27 + 7.39i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.830619882369048892991652409020, −9.039724008567746920369068107234, −8.181772681188123113590626667462, −7.35085461149932380982505756108, −6.99276964545003611898767648036, −5.49643995052538189355487938708, −4.62644936609091182875544150714, −4.08723601219882637699709604046, −2.61321101798368069192324880189, −0.32242968071794318824857439158,
0.815382014382314605043166285658, 2.52886471278543985345696762403, 3.73817959603087793365891493620, 4.31006936955463626061151295371, 5.68870312672567286846009845755, 6.82378217365940662220306316965, 7.69477096072419287580573089217, 8.455282940403414151994011505621, 8.642995582882031331478091297310, 10.54835537917556694638965515564