| L(s) = 1 | − 3-s + 9-s + 4·11-s + 2·13-s − 2·17-s − 2·19-s + 6·23-s − 27-s − 8·29-s − 8·31-s − 4·33-s + 6·37-s − 2·39-s + 2·41-s − 4·43-s − 6·47-s − 7·49-s + 2·51-s − 10·53-s + 2·57-s − 8·59-s + 2·61-s + 4·67-s − 6·69-s − 4·73-s − 4·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.458·19-s + 1.25·23-s − 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.696·33-s + 0.986·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s − 0.875·47-s − 49-s + 0.280·51-s − 1.37·53-s + 0.264·57-s − 1.04·59-s + 0.256·61-s + 0.488·67-s − 0.722·69-s − 0.468·73-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.23663191013185086724423531891, −6.53114031488991329329894158247, −6.14467240569553603053103102442, −5.29366626423860637659949594136, −4.59317791430158577405674947646, −3.85160618273778373125874059421, −3.20775324427209996725390935198, −1.93337942675812192479226379175, −1.25747991749413968753066308966, 0,
1.25747991749413968753066308966, 1.93337942675812192479226379175, 3.20775324427209996725390935198, 3.85160618273778373125874059421, 4.59317791430158577405674947646, 5.29366626423860637659949594136, 6.14467240569553603053103102442, 6.53114031488991329329894158247, 7.23663191013185086724423531891
