| L(s) = 1 | − 0.618·2-s − 1.61·4-s + 1.61·5-s + 3.23·7-s + 2.23·8-s − 1.00·10-s + 3.47·11-s − 3.61·13-s − 2.00·14-s + 1.85·16-s − 7.85·17-s + 6.09·19-s − 2.61·20-s − 2.14·22-s − 0.854·23-s − 2.38·25-s + 2.23·26-s − 5.23·28-s − 6.70·31-s − 5.61·32-s + 4.85·34-s + 5.23·35-s + 2.61·37-s − 3.76·38-s + 3.61·40-s + 5·41-s + 5·43-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.809·4-s + 0.723·5-s + 1.22·7-s + 0.790·8-s − 0.316·10-s + 1.04·11-s − 1.00·13-s − 0.534·14-s + 0.463·16-s − 1.90·17-s + 1.39·19-s − 0.585·20-s − 0.457·22-s − 0.178·23-s − 0.476·25-s + 0.438·26-s − 0.989·28-s − 1.20·31-s − 0.993·32-s + 0.832·34-s + 0.885·35-s + 0.430·37-s − 0.610·38-s + 0.572·40-s + 0.780·41-s + 0.762·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.726526219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.726526219\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 - 3.47T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 + 7.85T + 17T^{2} \) |
| 19 | \( 1 - 6.09T + 19T^{2} \) |
| 23 | \( 1 + 0.854T + 23T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 - 2.61T + 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 - 7T + 47T^{2} \) |
| 53 | \( 1 - 7.47T + 53T^{2} \) |
| 59 | \( 1 + 1.90T + 59T^{2} \) |
| 61 | \( 1 + 8.18T + 61T^{2} \) |
| 67 | \( 1 + 2.32T + 67T^{2} \) |
| 71 | \( 1 + 0.326T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 1.47T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−7.73243138727596098399831973799, −7.51815822684577078885118888079, −6.53675966543681827046654925409, −5.65884322567873753913792673289, −5.01078950481734477383498826802, −4.43001847139396220800579883197, −3.75763091463363578682196620836, −2.33591231849621454658436099596, −1.74080239561929345336631856954, −0.73574342979187529283701794100,
0.73574342979187529283701794100, 1.74080239561929345336631856954, 2.33591231849621454658436099596, 3.75763091463363578682196620836, 4.43001847139396220800579883197, 5.01078950481734477383498826802, 5.65884322567873753913792673289, 6.53675966543681827046654925409, 7.51815822684577078885118888079, 7.73243138727596098399831973799
