| L(s) = 1 | + 2·3-s − 2·5-s + 7-s + 9-s + 2·13-s − 4·15-s − 6·19-s + 2·21-s − 8·23-s − 25-s − 4·27-s − 6·31-s − 2·35-s − 10·37-s + 4·39-s + 12·41-s − 4·43-s − 2·45-s − 6·47-s + 49-s − 10·53-s − 12·57-s − 8·59-s + 12·61-s + 63-s − 4·65-s − 12·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 1.03·15-s − 1.37·19-s + 0.436·21-s − 1.66·23-s − 1/5·25-s − 0.769·27-s − 1.07·31-s − 0.338·35-s − 1.64·37-s + 0.640·39-s + 1.87·41-s − 0.609·43-s − 0.298·45-s − 0.875·47-s + 1/7·49-s − 1.37·53-s − 1.58·57-s − 1.04·59-s + 1.53·61-s + 0.125·63-s − 0.496·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.470828197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.470828197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.67962027179635, −14.06891016537994, −13.75303319089545, −13.12949726877938, −12.49644555153334, −12.13086894686499, −11.47564953151831, −10.87677098489618, −10.63132447372713, −9.667215083669694, −9.339046675639707, −8.576928366697067, −8.278398989871942, −7.886388279337553, −7.415063089597975, −6.606945201034037, −6.062591713779548, −5.360130652839226, −4.517186224810336, −3.935634744718093, −3.645639857303035, −2.942364597361550, −1.971029224753894, −1.810988999491984, −0.3734770943655791,
0.3734770943655791, 1.810988999491984, 1.971029224753894, 2.942364597361550, 3.645639857303035, 3.935634744718093, 4.517186224810336, 5.360130652839226, 6.062591713779548, 6.606945201034037, 7.415063089597975, 7.886388279337553, 8.278398989871942, 8.576928366697067, 9.339046675639707, 9.667215083669694, 10.63132447372713, 10.87677098489618, 11.47564953151831, 12.13086894686499, 12.49644555153334, 13.12949726877938, 13.75303319089545, 14.06891016537994, 14.67962027179635
