| L(s) = 1 | − 2.65·2-s + 5.03·4-s + 4.66·7-s − 8.05·8-s − 2.92·11-s − 0.896·13-s − 12.3·14-s + 11.3·16-s + 6.41·17-s + 8.64·19-s + 7.76·22-s − 5.17·23-s + 2.37·26-s + 23.4·28-s + 6.91·29-s + 5.33·31-s − 13.8·32-s − 17.0·34-s − 37-s − 22.9·38-s + 10.7·41-s − 5.70·43-s − 14.7·44-s + 13.7·46-s + 9.93·47-s + 14.7·49-s − 4.51·52-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 2.51·4-s + 1.76·7-s − 2.84·8-s − 0.882·11-s − 0.248·13-s − 3.30·14-s + 2.82·16-s + 1.55·17-s + 1.98·19-s + 1.65·22-s − 1.07·23-s + 0.466·26-s + 4.43·28-s + 1.28·29-s + 0.957·31-s − 2.45·32-s − 2.91·34-s − 0.164·37-s − 3.71·38-s + 1.67·41-s − 0.870·43-s − 2.22·44-s + 2.02·46-s + 1.44·47-s + 2.10·49-s − 0.626·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.278786711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.278786711\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 7 | \( 1 - 4.66T + 7T^{2} \) |
| 11 | \( 1 + 2.92T + 11T^{2} \) |
| 13 | \( 1 + 0.896T + 13T^{2} \) |
| 17 | \( 1 - 6.41T + 17T^{2} \) |
| 19 | \( 1 - 8.64T + 19T^{2} \) |
| 23 | \( 1 + 5.17T + 23T^{2} \) |
| 29 | \( 1 - 6.91T + 29T^{2} \) |
| 31 | \( 1 - 5.33T + 31T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 - 9.93T + 47T^{2} \) |
| 53 | \( 1 - 5.29T + 53T^{2} \) |
| 59 | \( 1 - 8.60T + 59T^{2} \) |
| 61 | \( 1 - 9.36T + 61T^{2} \) |
| 67 | \( 1 + 7.14T + 67T^{2} \) |
| 71 | \( 1 - 0.600T + 71T^{2} \) |
| 73 | \( 1 + 9.12T + 73T^{2} \) |
| 79 | \( 1 + 5.34T + 79T^{2} \) |
| 83 | \( 1 + 3.42T + 83T^{2} \) |
| 89 | \( 1 + 5.90T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.87864702487731126456225958750, −7.58778079320503586037180947293, −6.89038015611710949839942610759, −5.59713641176936000847615054398, −5.45614340349337834415866634685, −4.28846103745034323855474245006, −2.97240087558942301567877997895, −2.38001428767743452569241154581, −1.33069139943695423014703044106, −0.859763595207653099552745013496,
0.859763595207653099552745013496, 1.33069139943695423014703044106, 2.38001428767743452569241154581, 2.97240087558942301567877997895, 4.28846103745034323855474245006, 5.45614340349337834415866634685, 5.59713641176936000847615054398, 6.89038015611710949839942610759, 7.58778079320503586037180947293, 7.87864702487731126456225958750
