| L(s) = 1 | − 1.99·3-s − 1.57·5-s + 0.963·9-s − 5.45·11-s − 1.27·13-s + 3.14·15-s + 3.10·17-s − 1.80·19-s − 23-s − 2.51·25-s + 4.05·27-s + 8.86·29-s − 5.60·31-s + 10.8·33-s + 0.200·37-s + 2.54·39-s + 0.263·41-s + 11.2·43-s − 1.51·45-s − 8.45·47-s − 6.17·51-s + 13.4·53-s + 8.59·55-s + 3.58·57-s + 5.28·59-s − 1.84·61-s + 2.01·65-s + ⋯ |
| L(s) = 1 | − 1.14·3-s − 0.705·5-s + 0.321·9-s − 1.64·11-s − 0.354·13-s + 0.810·15-s + 0.752·17-s − 0.413·19-s − 0.208·23-s − 0.502·25-s + 0.780·27-s + 1.64·29-s − 1.00·31-s + 1.88·33-s + 0.0329·37-s + 0.406·39-s + 0.0412·41-s + 1.71·43-s − 0.226·45-s − 1.23·47-s − 0.864·51-s + 1.84·53-s + 1.15·55-s + 0.474·57-s + 0.688·59-s − 0.236·61-s + 0.249·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.99T + 3T^{2} \) |
| 5 | \( 1 + 1.57T + 5T^{2} \) |
| 11 | \( 1 + 5.45T + 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 + 1.80T + 19T^{2} \) |
| 29 | \( 1 - 8.86T + 29T^{2} \) |
| 31 | \( 1 + 5.60T + 31T^{2} \) |
| 37 | \( 1 - 0.200T + 37T^{2} \) |
| 41 | \( 1 - 0.263T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 8.45T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 5.28T + 59T^{2} \) |
| 61 | \( 1 + 1.84T + 61T^{2} \) |
| 67 | \( 1 + 3.22T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 - 2.95T + 73T^{2} \) |
| 79 | \( 1 - 9.32T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 6.66T + 97T^{2} \) |
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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
−7.39184471961972213092538076584, −6.70715398783561544568552551244, −5.82726007220035924476738053984, −5.40286474822047076081250894349, −4.76519132067657152591674199056, −4.01198457185804130545837151075, −3.02555032502306885626104380295, −2.26708294387441737783143431160, −0.824510011051550903311571464479, 0,
0.824510011051550903311571464479, 2.26708294387441737783143431160, 3.02555032502306885626104380295, 4.01198457185804130545837151075, 4.76519132067657152591674199056, 5.40286474822047076081250894349, 5.82726007220035924476738053984, 6.70715398783561544568552551244, 7.39184471961972213092538076584
