| L(s) = 1 | − 3-s − 3.70·5-s + 2.25·7-s + 9-s + 11-s + 13-s + 3.70·15-s + 2.65·17-s + 7.84·19-s − 2.25·21-s − 7.05·23-s + 8.74·25-s − 27-s + 5.44·29-s − 8.91·31-s − 33-s − 8.37·35-s − 0.791·37-s − 39-s + 4.25·41-s + 9.44·43-s − 3.70·45-s − 2.51·47-s − 1.89·49-s − 2.65·51-s + 7.82·53-s − 3.70·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.65·5-s + 0.854·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 0.957·15-s + 0.644·17-s + 1.79·19-s − 0.493·21-s − 1.47·23-s + 1.74·25-s − 0.192·27-s + 1.01·29-s − 1.60·31-s − 0.174·33-s − 1.41·35-s − 0.130·37-s − 0.160·39-s + 0.665·41-s + 1.44·43-s − 0.552·45-s − 0.367·47-s − 0.270·49-s − 0.371·51-s + 1.07·53-s − 0.499·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.276855863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.276855863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 - 2.25T + 7T^{2} \) |
| 17 | \( 1 - 2.65T + 17T^{2} \) |
| 19 | \( 1 - 7.84T + 19T^{2} \) |
| 23 | \( 1 + 7.05T + 23T^{2} \) |
| 29 | \( 1 - 5.44T + 29T^{2} \) |
| 31 | \( 1 + 8.91T + 31T^{2} \) |
| 37 | \( 1 + 0.791T + 37T^{2} \) |
| 41 | \( 1 - 4.25T + 41T^{2} \) |
| 43 | \( 1 - 9.44T + 43T^{2} \) |
| 47 | \( 1 + 2.51T + 47T^{2} \) |
| 53 | \( 1 - 7.82T + 53T^{2} \) |
| 59 | \( 1 + 7.41T + 59T^{2} \) |
| 61 | \( 1 - 6.62T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 2.79T + 71T^{2} \) |
| 73 | \( 1 - 6.53T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 7.41T + 83T^{2} \) |
| 89 | \( 1 - 4.66T + 89T^{2} \) |
| 97 | \( 1 - 3.41T + 97T^{2} \) |
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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
−7.80927913373223166979476641619, −7.47595501382089346028950225663, −6.71872836178099930201180468301, −5.65524951006287386306535156741, −5.18615675082768868801648102183, −4.18508501594817242387459919349, −3.87170865917067508571143920981, −2.91911133045217003606805333367, −1.51476828373388977875616990333, −0.64163223556179385548985614239,
0.64163223556179385548985614239, 1.51476828373388977875616990333, 2.91911133045217003606805333367, 3.87170865917067508571143920981, 4.18508501594817242387459919349, 5.18615675082768868801648102183, 5.65524951006287386306535156741, 6.71872836178099930201180468301, 7.47595501382089346028950225663, 7.80927913373223166979476641619
