| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s + 15-s + 4·19-s − 4·21-s + 4·23-s + 25-s − 27-s + 8·31-s − 4·35-s − 10·37-s − 12·41-s − 4·43-s − 45-s + 12·47-s + 9·49-s − 10·53-s − 4·57-s − 4·61-s + 4·63-s − 4·67-s − 4·69-s − 4·73-s − 75-s − 4·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.258·15-s + 0.917·19-s − 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.43·31-s − 0.676·35-s − 1.64·37-s − 1.87·41-s − 0.609·43-s − 0.149·45-s + 1.75·47-s + 9/7·49-s − 1.37·53-s − 0.529·57-s − 0.512·61-s + 0.503·63-s − 0.488·67-s − 0.481·69-s − 0.468·73-s − 0.115·75-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.63756169908698, −14.04651693122094, −13.66838664338907, −13.16603774186624, −12.21227099207087, −12.05276587938287, −11.61107467494957, −11.13148946645507, −10.52286843925318, −10.27368784054102, −9.445442748079598, −8.750310299231034, −8.443911044095484, −7.685472210735571, −7.441003054136236, −6.731750581351971, −6.189847804592194, −5.307427504680725, −4.975846319371971, −4.661052265392278, −3.792363582572239, −3.218447879169084, −2.358247658634207, −1.483582967126618, −1.081470622132554, 0,
1.081470622132554, 1.483582967126618, 2.358247658634207, 3.218447879169084, 3.792363582572239, 4.661052265392278, 4.975846319371971, 5.307427504680725, 6.189847804592194, 6.731750581351971, 7.441003054136236, 7.685472210735571, 8.443911044095484, 8.750310299231034, 9.445442748079598, 10.27368784054102, 10.52286843925318, 11.13148946645507, 11.61107467494957, 12.05276587938287, 12.21227099207087, 13.16603774186624, 13.66838664338907, 14.04651693122094, 14.63756169908698
