| L(s) = 1 | − 3·3-s − 3·5-s − 4·7-s + 6·9-s − 4·11-s − 5·13-s + 9·15-s − 6·17-s − 2·19-s + 12·21-s − 8·23-s + 4·25-s − 9·27-s − 9·29-s + 4·31-s + 12·33-s + 12·35-s − 37-s + 15·39-s − 10·41-s − 5·43-s − 18·45-s − 47-s + 9·49-s + 18·51-s − 4·53-s + 12·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s − 1.51·7-s + 2·9-s − 1.20·11-s − 1.38·13-s + 2.32·15-s − 1.45·17-s − 0.458·19-s + 2.61·21-s − 1.66·23-s + 4/5·25-s − 1.73·27-s − 1.67·29-s + 0.718·31-s + 2.08·33-s + 2.02·35-s − 0.164·37-s + 2.40·39-s − 1.56·41-s − 0.762·43-s − 2.68·45-s − 0.145·47-s + 9/7·49-s + 2.52·51-s − 0.549·53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−15.90930776747648, −15.71369235301292, −15.36253995719904, −14.75616135335327, −13.50345737083202, −13.29149899939537, −12.57821744945434, −12.19947195594755, −11.94074569207676, −11.18771797928288, −10.80604103966448, −10.15765412806717, −9.835131164102358, −9.126087139964090, −8.112274837352586, −7.712105676326037, −6.939104575792801, −6.704767546250983, −5.998337598644545, −5.406974334195293, −4.673583830524166, −4.294220509869643, −3.544410766732975, −2.720451563689213, −1.821966159135146, 0, 0, 0,
1.821966159135146, 2.720451563689213, 3.544410766732975, 4.294220509869643, 4.673583830524166, 5.406974334195293, 5.998337598644545, 6.704767546250983, 6.939104575792801, 7.712105676326037, 8.112274837352586, 9.126087139964090, 9.835131164102358, 10.15765412806717, 10.80604103966448, 11.18771797928288, 11.94074569207676, 12.19947195594755, 12.57821744945434, 13.29149899939537, 13.50345737083202, 14.75616135335327, 15.36253995719904, 15.71369235301292, 15.90930776747648
