| L(s) = 1 | + 3-s − 2·5-s − 7-s + 9-s + 4·11-s − 2·13-s − 2·15-s − 6·17-s − 4·19-s − 21-s + 23-s − 25-s + 27-s − 2·29-s + 8·31-s + 4·33-s + 2·35-s + 6·37-s − 2·39-s − 6·41-s + 4·43-s − 2·45-s + 8·47-s + 49-s − 6·51-s + 6·53-s − 8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.516·15-s − 1.45·17-s − 0.917·19-s − 0.218·21-s + 0.208·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s + 0.338·35-s + 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.655316589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.655316589\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−17.05297964852631, −16.44562804018155, −15.67541765886255, −15.18383627748458, −14.95548254130491, −14.06782784396406, −13.56786941333264, −12.94220916480844, −12.19488391994245, −11.81690391647423, −11.10949048578856, −10.46696137887687, −9.665197702746561, −9.038748868543925, −8.630958128255774, −7.855761961252678, −7.211399725498703, −6.570307326161091, −6.025004920252657, −4.650313960017298, −4.287438664468498, −3.629719966053854, −2.709292303889851, −1.922685395335419, −0.6079299525922811,
0.6079299525922811, 1.922685395335419, 2.709292303889851, 3.629719966053854, 4.287438664468498, 4.650313960017298, 6.025004920252657, 6.570307326161091, 7.211399725498703, 7.855761961252678, 8.630958128255774, 9.038748868543925, 9.665197702746561, 10.46696137887687, 11.10949048578856, 11.81690391647423, 12.19488391994245, 12.94220916480844, 13.56786941333264, 14.06782784396406, 14.95548254130491, 15.18383627748458, 15.67541765886255, 16.44562804018155, 17.05297964852631
