L(s) = 1 | − 2.22·2-s − 3-s + 2.96·4-s + 2.22·6-s − 2.15·8-s + 9-s + 1.77·11-s − 2.96·12-s − 4.19·13-s − 1.13·16-s + 0.322·17-s − 2.22·18-s + 7.66·19-s − 3.96·22-s + 4.97·23-s + 2.15·24-s + 9.34·26-s − 27-s − 8.08·29-s + 4.67·31-s + 6.83·32-s − 1.77·33-s − 0.719·34-s + 2.96·36-s − 10.2·37-s − 17.0·38-s + 4.19·39-s + ⋯ |
L(s) = 1 | − 1.57·2-s − 0.577·3-s + 1.48·4-s + 0.909·6-s − 0.760·8-s + 0.333·9-s + 0.536·11-s − 0.856·12-s − 1.16·13-s − 0.284·16-s + 0.0783·17-s − 0.525·18-s + 1.75·19-s − 0.845·22-s + 1.03·23-s + 0.439·24-s + 1.83·26-s − 0.192·27-s − 1.50·29-s + 0.838·31-s + 1.20·32-s − 0.309·33-s − 0.123·34-s + 0.494·36-s − 1.69·37-s − 2.77·38-s + 0.671·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6162713151\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6162713151\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 13 | \( 1 + 4.19T + 13T^{2} \) |
| 17 | \( 1 - 0.322T + 17T^{2} \) |
| 19 | \( 1 - 7.66T + 19T^{2} \) |
| 23 | \( 1 - 4.97T + 23T^{2} \) |
| 29 | \( 1 + 8.08T + 29T^{2} \) |
| 31 | \( 1 - 4.67T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 1.69T + 41T^{2} \) |
| 43 | \( 1 + 8.04T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 5.84T + 53T^{2} \) |
| 59 | \( 1 - 3.58T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 5.60T + 67T^{2} \) |
| 71 | \( 1 - 6.72T + 71T^{2} \) |
| 73 | \( 1 + 5.39T + 73T^{2} \) |
| 79 | \( 1 + 3.23T + 79T^{2} \) |
| 83 | \( 1 - 4.52T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 5.27T + 97T^{2} \) |
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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
−8.679172963276083146483474697030, −7.76765699722388714592053781345, −7.10517831365086334953161632667, −6.83629221826894751388254384544, −5.54418850692332192112466594911, −5.00828575282799187030948581198, −3.78720684413604251282972168328, −2.63510389393631930212506748816, −1.55488775365828281340573065627, −0.62997706228294034369770007968,
0.62997706228294034369770007968, 1.55488775365828281340573065627, 2.63510389393631930212506748816, 3.78720684413604251282972168328, 5.00828575282799187030948581198, 5.54418850692332192112466594911, 6.83629221826894751388254384544, 7.10517831365086334953161632667, 7.76765699722388714592053781345, 8.679172963276083146483474697030