| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 4·11-s − 12-s − 13-s − 14-s + 15-s + 16-s − 17-s − 18-s + 4·19-s − 20-s − 21-s + 4·22-s + 4·23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s + 0.852·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.056384836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.056384836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.71749736547776, −14.18467928610340, −13.43340032748493, −13.02545708240323, −12.43946309912268, −11.78483232698238, −11.55074472208312, −10.93850771156190, −10.39072851315662, −10.12322535433247, −9.334860683337367, −8.875911850244043, −8.085681624402916, −7.824003102502407, −7.234436723759576, −6.735745670915338, −6.023255916315954, −5.340027807156157, −4.916702296476009, −4.318239869998494, −3.355543785591290, −2.771011882280273, −2.081175254809904, −1.097959867097588, −0.4940707564867200,
0.4940707564867200, 1.097959867097588, 2.081175254809904, 2.771011882280273, 3.355543785591290, 4.318239869998494, 4.916702296476009, 5.340027807156157, 6.023255916315954, 6.735745670915338, 7.234436723759576, 7.824003102502407, 8.085681624402916, 8.875911850244043, 9.334860683337367, 10.12322535433247, 10.39072851315662, 10.93850771156190, 11.55074472208312, 11.78483232698238, 12.43946309912268, 13.02545708240323, 13.43340032748493, 14.18467928610340, 14.71749736547776
