| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 5·11-s + 2·13-s + 15-s − 17-s + 19-s + 21-s + 2·23-s + 25-s − 27-s + 9·29-s + 10·31-s + 5·33-s + 35-s + 9·37-s − 2·39-s − 3·41-s − 4·43-s − 45-s − 3·47-s − 6·49-s + 51-s − 5·53-s + 5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 0.554·13-s + 0.258·15-s − 0.242·17-s + 0.229·19-s + 0.218·21-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 1.67·29-s + 1.79·31-s + 0.870·33-s + 0.169·35-s + 1.47·37-s − 0.320·39-s − 0.468·41-s − 0.609·43-s − 0.149·45-s − 0.437·47-s − 6/7·49-s + 0.140·51-s − 0.686·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.058316345728194951676131585577, −7.38525581334719424926303173150, −6.41914210712003652815618845944, −6.03006994335128848445182390135, −4.81531029953622689966459912048, −4.62308614710142819848654932510, −3.24909628018459333645248617559, −2.67969482382916406179778025610, −1.17765785423550573877334943927, 0,
1.17765785423550573877334943927, 2.67969482382916406179778025610, 3.24909628018459333645248617559, 4.62308614710142819848654932510, 4.81531029953622689966459912048, 6.03006994335128848445182390135, 6.41914210712003652815618845944, 7.38525581334719424926303173150, 8.058316345728194951676131585577
