| L(s) = 1 | − 3-s − 3·7-s + 9-s − 11-s + 4·13-s − 17-s − 6·19-s + 3·21-s − 2·23-s − 27-s + 5·29-s + 33-s − 2·37-s − 4·39-s + 9·41-s − 2·43-s − 7·47-s + 2·49-s + 51-s − 9·53-s + 6·57-s − 3·59-s − 10·61-s − 3·63-s + 13·67-s + 2·69-s − 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.242·17-s − 1.37·19-s + 0.654·21-s − 0.417·23-s − 0.192·27-s + 0.928·29-s + 0.174·33-s − 0.328·37-s − 0.640·39-s + 1.40·41-s − 0.304·43-s − 1.02·47-s + 2/7·49-s + 0.140·51-s − 1.23·53-s + 0.794·57-s − 0.390·59-s − 1.28·61-s − 0.377·63-s + 1.58·67-s + 0.240·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.05845933910864, −12.59055140066435, −12.51724724748428, −11.81604490867877, −11.08961229655406, −11.01312884914819, −10.38700891691145, −10.05284291244724, −9.396281003956383, −9.119742310803650, −8.375372455197015, −8.106208677357775, −7.476482852108535, −6.676862961320055, −6.459654373999108, −6.161644226422548, −5.623087143524982, −4.870789938479249, −4.473747130593059, −3.815871985274626, −3.400038781882827, −2.757272435787981, −2.110542153304219, −1.431910541482327, −0.6276291452595290, 0,
0.6276291452595290, 1.431910541482327, 2.110542153304219, 2.757272435787981, 3.400038781882827, 3.815871985274626, 4.473747130593059, 4.870789938479249, 5.623087143524982, 6.161644226422548, 6.459654373999108, 6.676862961320055, 7.476482852108535, 8.106208677357775, 8.375372455197015, 9.119742310803650, 9.396281003956383, 10.05284291244724, 10.38700891691145, 11.01312884914819, 11.08961229655406, 11.81604490867877, 12.51724724748428, 12.59055140066435, 13.05845933910864
