| L(s) = 1 | − 0.857·3-s + 5-s − 2.23·7-s − 2.26·9-s + 3.21·11-s + 3.25·13-s − 0.857·15-s + 0.870·17-s − 5.02·19-s + 1.91·21-s − 8.46·23-s + 25-s + 4.51·27-s + 29-s − 0.0467·31-s − 2.75·33-s − 2.23·35-s + 11.1·37-s − 2.78·39-s − 2.47·41-s − 2.87·43-s − 2.26·45-s − 2.20·47-s − 1.99·49-s − 0.746·51-s − 11.7·53-s + 3.21·55-s + ⋯ |
| L(s) = 1 | − 0.495·3-s + 0.447·5-s − 0.845·7-s − 0.754·9-s + 0.970·11-s + 0.901·13-s − 0.221·15-s + 0.211·17-s − 1.15·19-s + 0.418·21-s − 1.76·23-s + 0.200·25-s + 0.869·27-s + 0.185·29-s − 0.00839·31-s − 0.480·33-s − 0.378·35-s + 1.82·37-s − 0.446·39-s − 0.386·41-s − 0.437·43-s − 0.337·45-s − 0.321·47-s − 0.284·49-s − 0.104·51-s − 1.61·53-s + 0.433·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.857T + 3T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 - 3.21T + 11T^{2} \) |
| 13 | \( 1 - 3.25T + 13T^{2} \) |
| 17 | \( 1 - 0.870T + 17T^{2} \) |
| 19 | \( 1 + 5.02T + 19T^{2} \) |
| 23 | \( 1 + 8.46T + 23T^{2} \) |
| 31 | \( 1 + 0.0467T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 + 2.87T + 43T^{2} \) |
| 47 | \( 1 + 2.20T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 7.68T + 59T^{2} \) |
| 61 | \( 1 + 4.99T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 3.43T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 6.44T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 7.16T + 97T^{2} \) |
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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
−8.646105262784239625125037835388, −7.996712059734354470346653712296, −6.69155100702590672734694161739, −6.15002442477047797403580457019, −5.86854034917099693709359681529, −4.54776113601425447121358042493, −3.72213409492117670013358292200, −2.73878388439611038319977209540, −1.49682472527592778986330103374, 0,
1.49682472527592778986330103374, 2.73878388439611038319977209540, 3.72213409492117670013358292200, 4.54776113601425447121358042493, 5.86854034917099693709359681529, 6.15002442477047797403580457019, 6.69155100702590672734694161739, 7.996712059734354470346653712296, 8.646105262784239625125037835388
