| L(s) = 1 | + 2-s − 2.52·3-s + 4-s − 2.27·5-s − 2.52·6-s + 4.02·7-s + 8-s + 3.38·9-s − 2.27·10-s − 4.07·11-s − 2.52·12-s − 1.05·13-s + 4.02·14-s + 5.74·15-s + 16-s + 3.99·17-s + 3.38·18-s − 0.689·19-s − 2.27·20-s − 10.1·21-s − 4.07·22-s − 23-s − 2.52·24-s + 0.174·25-s − 1.05·26-s − 0.968·27-s + 4.02·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.45·3-s + 0.5·4-s − 1.01·5-s − 1.03·6-s + 1.51·7-s + 0.353·8-s + 1.12·9-s − 0.719·10-s − 1.22·11-s − 0.729·12-s − 0.291·13-s + 1.07·14-s + 1.48·15-s + 0.250·16-s + 0.968·17-s + 0.797·18-s − 0.158·19-s − 0.508·20-s − 2.21·21-s − 0.868·22-s − 0.208·23-s − 0.515·24-s + 0.0349·25-s − 0.206·26-s − 0.186·27-s + 0.759·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.373769101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.373769101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2.52T + 3T^{2} \) |
| 5 | \( 1 + 2.27T + 5T^{2} \) |
| 7 | \( 1 - 4.02T + 7T^{2} \) |
| 11 | \( 1 + 4.07T + 11T^{2} \) |
| 13 | \( 1 + 1.05T + 13T^{2} \) |
| 17 | \( 1 - 3.99T + 17T^{2} \) |
| 19 | \( 1 + 0.689T + 19T^{2} \) |
| 31 | \( 1 - 4.06T + 31T^{2} \) |
| 37 | \( 1 + 1.35T + 37T^{2} \) |
| 41 | \( 1 - 5.36T + 41T^{2} \) |
| 43 | \( 1 - 9.08T + 43T^{2} \) |
| 47 | \( 1 - 7.52T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 8.29T + 59T^{2} \) |
| 61 | \( 1 + 0.914T + 61T^{2} \) |
| 67 | \( 1 - 6.25T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 8.81T + 73T^{2} \) |
| 79 | \( 1 - 2.41T + 79T^{2} \) |
| 83 | \( 1 + 3.22T + 83T^{2} \) |
| 89 | \( 1 - 5.58T + 89T^{2} \) |
| 97 | \( 1 - 1.81T + 97T^{2} \) |
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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
−10.11469563349646502239496664121, −8.484148423808856497471625214642, −7.67628682358905132942291257296, −7.29392492521117621567561200173, −5.93116803850332299200178450222, −5.34785933081547636711142626867, −4.70524092759240448178383721491, −3.94862115773677661317502471034, −2.42191710818900015975923991608, −0.836543118505236410779992407405,
0.836543118505236410779992407405, 2.42191710818900015975923991608, 3.94862115773677661317502471034, 4.70524092759240448178383721491, 5.34785933081547636711142626867, 5.93116803850332299200178450222, 7.29392492521117621567561200173, 7.67628682358905132942291257296, 8.484148423808856497471625214642, 10.11469563349646502239496664121
