| L(s) = 1 | + 3·3-s − 4·5-s + 3·7-s + 6·9-s + 3·11-s − 7·13-s − 12·15-s − 2·17-s + 2·19-s + 9·21-s + 3·23-s + 11·25-s + 9·27-s − 8·29-s + 10·31-s + 9·33-s − 12·35-s − 8·37-s − 21·39-s − 6·41-s + 11·43-s − 24·45-s − 3·47-s + 2·49-s − 6·51-s − 4·53-s − 12·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.78·5-s + 1.13·7-s + 2·9-s + 0.904·11-s − 1.94·13-s − 3.09·15-s − 0.485·17-s + 0.458·19-s + 1.96·21-s + 0.625·23-s + 11/5·25-s + 1.73·27-s − 1.48·29-s + 1.79·31-s + 1.56·33-s − 2.02·35-s − 1.31·37-s − 3.36·39-s − 0.937·41-s + 1.67·43-s − 3.57·45-s − 0.437·47-s + 2/7·49-s − 0.840·51-s − 0.549·53-s − 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.529669064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.529669064\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−15.13345102482653, −14.53963339172431, −14.13640657556507, −13.84615055154150, −12.72819931364190, −12.53775545830214, −11.85964569454703, −11.40973897294825, −10.96133359843595, −10.01375037791742, −9.535842710323085, −8.971020262156000, −8.336641277556577, −8.117616926947180, −7.464208782605241, −7.190344037458888, −6.662973361174772, −5.064785940282770, −4.863797239183069, −4.064469848451229, −3.763081368381461, −2.990620535744946, −2.374448107091255, −1.640773428499031, −0.6456066512676714,
0.6456066512676714, 1.640773428499031, 2.374448107091255, 2.990620535744946, 3.763081368381461, 4.064469848451229, 4.863797239183069, 5.064785940282770, 6.662973361174772, 7.190344037458888, 7.464208782605241, 8.117616926947180, 8.336641277556577, 8.971020262156000, 9.535842710323085, 10.01375037791742, 10.96133359843595, 11.40973897294825, 11.85964569454703, 12.53775545830214, 12.72819931364190, 13.84615055154150, 14.13640657556507, 14.53963339172431, 15.13345102482653
