| L(s) = 1 | − 1.93·2-s − 1.61·3-s + 1.75·4-s − 0.710·5-s + 3.12·6-s − 1.51·7-s + 0.466·8-s − 0.394·9-s + 1.37·10-s + 4.83·11-s − 2.84·12-s − 3.42·13-s + 2.94·14-s + 1.14·15-s − 4.42·16-s + 2.25·17-s + 0.764·18-s + 19-s − 1.24·20-s + 2.44·21-s − 9.38·22-s + 5.02·23-s − 0.752·24-s − 4.49·25-s + 6.64·26-s + 5.47·27-s − 2.66·28-s + ⋯ |
| L(s) = 1 | − 1.37·2-s − 0.931·3-s + 0.879·4-s − 0.317·5-s + 1.27·6-s − 0.573·7-s + 0.164·8-s − 0.131·9-s + 0.435·10-s + 1.45·11-s − 0.819·12-s − 0.950·13-s + 0.785·14-s + 0.295·15-s − 1.10·16-s + 0.546·17-s + 0.180·18-s + 0.229·19-s − 0.279·20-s + 0.534·21-s − 1.99·22-s + 1.04·23-s − 0.153·24-s − 0.899·25-s + 1.30·26-s + 1.05·27-s − 0.504·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 + 0.710T + 5T^{2} \) |
| 7 | \( 1 + 1.51T + 7T^{2} \) |
| 11 | \( 1 - 4.83T + 11T^{2} \) |
| 13 | \( 1 + 3.42T + 13T^{2} \) |
| 17 | \( 1 - 2.25T + 17T^{2} \) |
| 23 | \( 1 - 5.02T + 23T^{2} \) |
| 29 | \( 1 + 9.09T + 29T^{2} \) |
| 31 | \( 1 + 7.95T + 31T^{2} \) |
| 37 | \( 1 - 0.118T + 37T^{2} \) |
| 41 | \( 1 - 4.81T + 41T^{2} \) |
| 43 | \( 1 - 7.87T + 43T^{2} \) |
| 47 | \( 1 + 4.37T + 47T^{2} \) |
| 53 | \( 1 + 2.80T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 5.59T + 61T^{2} \) |
| 67 | \( 1 + 4.65T + 67T^{2} \) |
| 71 | \( 1 - 4.73T + 71T^{2} \) |
| 73 | \( 1 - 8.30T + 73T^{2} \) |
| 79 | \( 1 + 1.73T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 0.175T + 89T^{2} \) |
| 97 | \( 1 + 6.23T + 97T^{2} \) |
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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
−8.131506772529825068859214661805, −7.24225888569126239546067052618, −6.96930234888979616006813471134, −5.98130882257511234827638695343, −5.30741859440847368058207743171, −4.26021067674264870687868149456, −3.38501216284795209806994731869, −2.05464012944551600904610968814, −0.941753162761719951581463022304, 0,
0.941753162761719951581463022304, 2.05464012944551600904610968814, 3.38501216284795209806994731869, 4.26021067674264870687868149456, 5.30741859440847368058207743171, 5.98130882257511234827638695343, 6.96930234888979616006813471134, 7.24225888569126239546067052618, 8.131506772529825068859214661805
