| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 2·11-s − 5·13-s − 15-s + 17-s − 2·19-s − 21-s + 23-s + 25-s + 27-s + 8·29-s − 31-s − 2·33-s + 35-s − 3·37-s − 5·39-s − 7·41-s − 45-s + 47-s + 49-s + 51-s − 8·53-s + 2·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.38·13-s − 0.258·15-s + 0.242·17-s − 0.458·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.48·29-s − 0.179·31-s − 0.348·33-s + 0.169·35-s − 0.493·37-s − 0.800·39-s − 1.09·41-s − 0.149·45-s + 0.145·47-s + 1/7·49-s + 0.140·51-s − 1.09·53-s + 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.277633564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.277633564\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.18696217108943, −14.79687426494755, −14.07447558877905, −13.72936120613912, −13.04153663840477, −12.44252985538222, −12.20242104369694, −11.56607308590295, −10.70303683016858, −10.31548410735084, −9.834628760383561, −9.160192772657764, −8.667197725362783, −7.933092922274923, −7.647231551145510, −6.909419920499113, −6.477014084667906, −5.575181693099584, −4.810403682166656, −4.524974041983642, −3.543077389029834, −2.989140156234428, −2.444901152990303, −1.583409890235580, −0.4090049641261567,
0.4090049641261567, 1.583409890235580, 2.444901152990303, 2.989140156234428, 3.543077389029834, 4.524974041983642, 4.810403682166656, 5.575181693099584, 6.477014084667906, 6.909419920499113, 7.647231551145510, 7.933092922274923, 8.667197725362783, 9.160192772657764, 9.834628760383561, 10.31548410735084, 10.70303683016858, 11.56607308590295, 12.20242104369694, 12.44252985538222, 13.04153663840477, 13.72936120613912, 14.07447558877905, 14.79687426494755, 15.18696217108943
