| L(s) = 1 | − 0.762·2-s + 3-s − 1.41·4-s − 1.72·5-s − 0.762·6-s + 1.91·7-s + 2.60·8-s + 9-s + 1.31·10-s − 11-s − 1.41·12-s + 2.75·13-s − 1.46·14-s − 1.72·15-s + 0.847·16-s + 6.48·17-s − 0.762·18-s − 3.45·19-s + 2.44·20-s + 1.91·21-s + 0.762·22-s − 2.31·23-s + 2.60·24-s − 2.01·25-s − 2.09·26-s + 27-s − 2.71·28-s + ⋯ |
| L(s) = 1 | − 0.539·2-s + 0.577·3-s − 0.709·4-s − 0.772·5-s − 0.311·6-s + 0.724·7-s + 0.921·8-s + 0.333·9-s + 0.416·10-s − 0.301·11-s − 0.409·12-s + 0.763·13-s − 0.390·14-s − 0.445·15-s + 0.211·16-s + 1.57·17-s − 0.179·18-s − 0.792·19-s + 0.547·20-s + 0.418·21-s + 0.162·22-s − 0.482·23-s + 0.532·24-s − 0.403·25-s − 0.411·26-s + 0.192·27-s − 0.513·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.272578829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.272578829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.762T + 2T^{2} \) |
| 5 | \( 1 + 1.72T + 5T^{2} \) |
| 7 | \( 1 - 1.91T + 7T^{2} \) |
| 13 | \( 1 - 2.75T + 13T^{2} \) |
| 17 | \( 1 - 6.48T + 17T^{2} \) |
| 19 | \( 1 + 3.45T + 19T^{2} \) |
| 23 | \( 1 + 2.31T + 23T^{2} \) |
| 29 | \( 1 - 3.30T + 29T^{2} \) |
| 31 | \( 1 + 8.07T + 31T^{2} \) |
| 37 | \( 1 + 6.14T + 37T^{2} \) |
| 41 | \( 1 - 5.53T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 - 3.58T + 47T^{2} \) |
| 53 | \( 1 - 0.104T + 53T^{2} \) |
| 59 | \( 1 - 8.73T + 59T^{2} \) |
| 67 | \( 1 + 1.77T + 67T^{2} \) |
| 71 | \( 1 - 9.47T + 71T^{2} \) |
| 73 | \( 1 - 6.67T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 7.45T + 83T^{2} \) |
| 89 | \( 1 - 0.855T + 89T^{2} \) |
| 97 | \( 1 - 7.45T + 97T^{2} \) |
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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
−9.000368208474893185830781652095, −8.285352622702631619230298892725, −7.87070957444535155290141837960, −7.29527427830336743660855049117, −5.88658817372497495390598534210, −5.01371570552022584436295137619, −4.03041706043210383724889866862, −3.55111428530885494434388202518, −2.01879060058742614971013386242, −0.827243222233260297349724142485,
0.827243222233260297349724142485, 2.01879060058742614971013386242, 3.55111428530885494434388202518, 4.03041706043210383724889866862, 5.01371570552022584436295137619, 5.88658817372497495390598534210, 7.29527427830336743660855049117, 7.87070957444535155290141837960, 8.285352622702631619230298892725, 9.000368208474893185830781652095
