L(s) = 1 | − 2.60·3-s − 2.76i·7-s + 3.76·9-s + 2.16i·11-s + (0.167 + 3.60i)13-s − 5.03i·19-s + 7.20i·21-s + 4.93·23-s − 2.00·27-s − 4.43·29-s + 3.37i·31-s − 5.63i·33-s + 3.97i·37-s + (−0.434 − 9.37i)39-s + 1.66i·41-s + ⋯ |
L(s) = 1 | − 1.50·3-s − 1.04i·7-s + 1.25·9-s + 0.653i·11-s + (0.0463 + 0.998i)13-s − 1.15i·19-s + 1.57i·21-s + 1.02·23-s − 0.384·27-s − 0.823·29-s + 0.605i·31-s − 0.981i·33-s + 0.653i·37-s + (−0.0695 − 1.50i)39-s + 0.260i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0463 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0463 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6857197139\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6857197139\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-0.167 - 3.60i)T \) |
good | 3 | \( 1 + 2.60T + 3T^{2} \) |
| 7 | \( 1 + 2.76iT - 7T^{2} \) |
| 11 | \( 1 - 2.16iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.03iT - 19T^{2} \) |
| 23 | \( 1 - 4.93T + 23T^{2} \) |
| 29 | \( 1 + 4.43T + 29T^{2} \) |
| 31 | \( 1 - 3.37iT - 31T^{2} \) |
| 37 | \( 1 - 3.97iT - 37T^{2} \) |
| 41 | \( 1 - 1.66iT - 41T^{2} \) |
| 43 | \( 1 + 9.80T + 43T^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 8.16iT - 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 7.10iT - 67T^{2} \) |
| 71 | \( 1 + 16.1iT - 71T^{2} \) |
| 73 | \( 1 + 15.9iT - 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 3.97iT - 83T^{2} \) |
| 89 | \( 1 + 7.94iT - 89T^{2} \) |
| 97 | \( 1 - 0.462iT - 97T^{2} \) |
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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
−9.642184419520382414608497891430, −8.783099166952189867893287076445, −7.39356131549609732082240457358, −6.90609008660132715534991073415, −6.29733042088359346367000091189, −4.96542294962923483192357784545, −4.72138056923647007382608059024, −3.50944830508777136577020085662, −1.74196641439494873265236140574, −0.42693358579448986931926694693,
1.06992074157931941490085426819, 2.66412814623519845847892693372, 3.90183426130113015537762519646, 5.23201020907555316391335620239, 5.62736236067938045970991999757, 6.20503521289655367006572064236, 7.26875395798551977679410883963, 8.240461659742408722829007609335, 9.063583823345988727344261012536, 10.06957722621729816214669322990