L(s) = 1 | + 3-s + 5-s + 9-s + 11-s − 2·13-s + 15-s + 6·17-s + 8·19-s + 6·23-s + 25-s + 27-s + 6·29-s + 2·31-s + 33-s + 2·37-s − 2·39-s − 8·43-s + 45-s − 12·47-s + 6·51-s + 6·53-s + 55-s + 8·57-s + 6·59-s − 8·61-s − 2·65-s − 2·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.258·15-s + 1.45·17-s + 1.83·19-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.174·33-s + 0.328·37-s − 0.320·39-s − 1.21·43-s + 0.149·45-s − 1.75·47-s + 0.840·51-s + 0.824·53-s + 0.134·55-s + 1.05·57-s + 0.781·59-s − 1.02·61-s − 0.248·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.262524776\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.262524776\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.53172672843550, −13.17643857718694, −12.45966972899032, −12.09779918502600, −11.67359689318608, −11.13265749719819, −10.39619800659046, −9.950975513062318, −9.645867217806937, −9.258957733518704, −8.571738781512043, −8.069948627335927, −7.670070659037287, −6.976746763088872, −6.761894614883661, −5.947551551924928, −5.326744345489767, −5.014180486369211, −4.422391073529918, −3.509814098668904, −3.111692428172692, −2.790449078165450, −1.855107589359461, −1.223158283145512, −0.7396908571337522,
0.7396908571337522, 1.223158283145512, 1.855107589359461, 2.790449078165450, 3.111692428172692, 3.509814098668904, 4.422391073529918, 5.014180486369211, 5.326744345489767, 5.947551551924928, 6.761894614883661, 6.976746763088872, 7.670070659037287, 8.069948627335927, 8.571738781512043, 9.258957733518704, 9.645867217806937, 9.950975513062318, 10.39619800659046, 11.13265749719819, 11.67359689318608, 12.09779918502600, 12.45966972899032, 13.17643857718694, 13.53172672843550