| L(s) = 1 | − 4·7-s − 3·9-s − 2·11-s + 2·13-s + 2·17-s + 2·19-s + 23-s + 2·29-s + 4·37-s + 6·41-s + 10·43-s + 9·49-s + 4·53-s − 12·59-s − 8·61-s + 12·63-s − 10·67-s − 6·73-s + 8·77-s + 12·79-s + 9·81-s + 14·83-s − 6·89-s − 8·91-s − 6·97-s + 6·99-s − 10·101-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 9-s − 0.603·11-s + 0.554·13-s + 0.485·17-s + 0.458·19-s + 0.208·23-s + 0.371·29-s + 0.657·37-s + 0.937·41-s + 1.52·43-s + 9/7·49-s + 0.549·53-s − 1.56·59-s − 1.02·61-s + 1.51·63-s − 1.22·67-s − 0.702·73-s + 0.911·77-s + 1.35·79-s + 81-s + 1.53·83-s − 0.635·89-s − 0.838·91-s − 0.609·97-s + 0.603·99-s − 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.46857534535789493800015604537, −6.54208983440276677800681635815, −5.97575849731990772435341906727, −5.58150831857899769559676699716, −4.57207872568795006212767634985, −3.64388322554704982328156851617, −2.99590359077282489191647819997, −2.52129486812440468870322717219, −1.02894047304490763031597515709, 0,
1.02894047304490763031597515709, 2.52129486812440468870322717219, 2.99590359077282489191647819997, 3.64388322554704982328156851617, 4.57207872568795006212767634985, 5.58150831857899769559676699716, 5.97575849731990772435341906727, 6.54208983440276677800681635815, 7.46857534535789493800015604537
