L(s) = 1 | + (1.29 + 0.576i)2-s + (1.33 + 1.48i)4-s − 1.97·7-s + (0.867 + 2.69i)8-s + 1.43i·11-s + 0.241i·13-s + (−2.55 − 1.13i)14-s + (−0.430 + 3.97i)16-s + 7.38·17-s + 3.04i·19-s + (−0.824 + 1.84i)22-s − 0.874·23-s + (−0.139 + 0.311i)26-s + (−2.64 − 2.94i)28-s + 9.07i·29-s + ⋯ |
L(s) = 1 | + (0.913 + 0.407i)2-s + (0.667 + 0.744i)4-s − 0.747·7-s + (0.306 + 0.951i)8-s + 0.431i·11-s + 0.0669i·13-s + (−0.682 − 0.304i)14-s + (−0.107 + 0.994i)16-s + 1.79·17-s + 0.697i·19-s + (−0.175 + 0.393i)22-s − 0.182·23-s + (−0.0272 + 0.0611i)26-s + (−0.499 − 0.556i)28-s + 1.68i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.664658748\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.664658748\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.29 - 0.576i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.97T + 7T^{2} \) |
| 11 | \( 1 - 1.43iT - 11T^{2} \) |
| 13 | \( 1 - 0.241iT - 13T^{2} \) |
| 17 | \( 1 - 7.38T + 17T^{2} \) |
| 19 | \( 1 - 3.04iT - 19T^{2} \) |
| 23 | \( 1 + 0.874T + 23T^{2} \) |
| 29 | \( 1 - 9.07iT - 29T^{2} \) |
| 31 | \( 1 + 7.44T + 31T^{2} \) |
| 37 | \( 1 + 8.81iT - 37T^{2} \) |
| 41 | \( 1 - 1.91T + 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 3.34T + 47T^{2} \) |
| 53 | \( 1 - 9.20iT - 53T^{2} \) |
| 59 | \( 1 - 6.43iT - 59T^{2} \) |
| 61 | \( 1 - 4.57iT - 61T^{2} \) |
| 67 | \( 1 + 4.86iT - 67T^{2} \) |
| 71 | \( 1 - 8.21T + 71T^{2} \) |
| 73 | \( 1 + 4.12T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 12.3iT - 83T^{2} \) |
| 89 | \( 1 - 8.08T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−9.501891984045789188852046564101, −8.622656845024548809913016309636, −7.47969786769726957974710363462, −7.29551750830680005985243216325, −6.03664082860279428858541067469, −5.65668582386620459032871391465, −4.60996824722145180714865394472, −3.60319906397029634411651089290, −3.01610735754576269658219980119, −1.63017324079792585932430427446,
0.72468132214361402131071724136, 2.17670013114108236161866432605, 3.26004420680977716841618284171, 3.78175245245725260656146012182, 5.00215697978040786708469019702, 5.71428686611222272637055354452, 6.42687295396130342521325895387, 7.28227829400118561611267903204, 8.145880327954646847506036911665, 9.336375841050390365995138851135