| L(s) = 1 | − 3-s + 9-s − 4·11-s − 6·13-s + 2·17-s + 4·23-s − 27-s + 2·29-s − 8·31-s + 4·33-s − 2·37-s + 6·39-s + 10·41-s + 43-s − 4·47-s − 7·49-s − 2·51-s + 14·53-s + 4·59-s − 2·61-s − 4·67-s − 4·69-s − 14·73-s − 8·79-s + 81-s + 8·83-s − 2·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 0.485·17-s + 0.834·23-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.696·33-s − 0.328·37-s + 0.960·39-s + 1.56·41-s + 0.152·43-s − 0.583·47-s − 49-s − 0.280·51-s + 1.92·53-s + 0.520·59-s − 0.256·61-s − 0.488·67-s − 0.481·69-s − 1.63·73-s − 0.900·79-s + 1/9·81-s + 0.878·83-s − 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.86920704938842, −14.32107406239994, −13.68579918628535, −12.98184271457956, −12.66388906438878, −12.37346016781698, −11.47711612181368, −11.34042646453365, −10.46386500672968, −10.21172588722948, −9.723500062446415, −9.040750196844141, −8.514042920758942, −7.612432583678788, −7.408285245316778, −7.002820332883651, −6.021282616000825, −5.640224068173809, −4.923241252232568, −4.771619521612256, −3.850283960524540, −3.027390508937102, −2.510339875303944, −1.798805762203548, −0.7532976505452261, 0,
0.7532976505452261, 1.798805762203548, 2.510339875303944, 3.027390508937102, 3.850283960524540, 4.771619521612256, 4.923241252232568, 5.640224068173809, 6.021282616000825, 7.002820332883651, 7.408285245316778, 7.612432583678788, 8.514042920758942, 9.040750196844141, 9.723500062446415, 10.21172588722948, 10.46386500672968, 11.34042646453365, 11.47711612181368, 12.37346016781698, 12.66388906438878, 12.98184271457956, 13.68579918628535, 14.32107406239994, 14.86920704938842
