| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 5·11-s − 7·13-s − 15-s − 3·17-s + 7·19-s − 4·21-s + 3·23-s + 25-s − 27-s + 29-s + 5·31-s − 5·33-s + 4·35-s − 9·37-s + 7·39-s − 4·43-s + 45-s + 4·47-s + 9·49-s + 3·51-s − 14·53-s + 5·55-s − 7·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.50·11-s − 1.94·13-s − 0.258·15-s − 0.727·17-s + 1.60·19-s − 0.872·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s + 0.898·31-s − 0.870·33-s + 0.676·35-s − 1.47·37-s + 1.12·39-s − 0.609·43-s + 0.149·45-s + 0.583·47-s + 9/7·49-s + 0.420·51-s − 1.92·53-s + 0.674·55-s − 0.927·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.92632868789609, −14.48638708039876, −13.99212290210736, −13.79520940870031, −12.83214008580573, −12.20363157784451, −11.89379451436260, −11.48861723258729, −11.00491393687642, −10.28503267640431, −9.785098649413981, −9.258509692722911, −8.789062241198644, −8.004531562498934, −7.434490902079328, −6.939415949694153, −6.491314281776690, −5.572858516001975, −5.113452704735066, −4.683602042141806, −4.205117489202094, −3.160663937286785, −2.452334032194050, −1.485339881706781, −1.311133281663739, 0,
1.311133281663739, 1.485339881706781, 2.452334032194050, 3.160663937286785, 4.205117489202094, 4.683602042141806, 5.113452704735066, 5.572858516001975, 6.491314281776690, 6.939415949694153, 7.434490902079328, 8.004531562498934, 8.789062241198644, 9.258509692722911, 9.785098649413981, 10.28503267640431, 11.00491393687642, 11.48861723258729, 11.89379451436260, 12.20363157784451, 12.83214008580573, 13.79520940870031, 13.99212290210736, 14.48638708039876, 14.92632868789609
