| L(s) = 1 | − 5-s + 2·7-s − 3·9-s + 4·11-s − 2·13-s − 2·17-s + 6·23-s + 25-s − 2·29-s − 8·31-s − 2·35-s − 10·37-s − 6·41-s + 4·43-s + 3·45-s + 12·47-s − 3·49-s − 14·53-s − 4·55-s + 4·61-s − 6·63-s + 2·65-s − 8·67-s + 12·71-s + 16·73-s + 8·77-s − 4·79-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 9-s + 1.20·11-s − 0.554·13-s − 0.485·17-s + 1.25·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.338·35-s − 1.64·37-s − 0.937·41-s + 0.609·43-s + 0.447·45-s + 1.75·47-s − 3/7·49-s − 1.92·53-s − 0.539·55-s + 0.512·61-s − 0.755·63-s + 0.248·65-s − 0.977·67-s + 1.42·71-s + 1.87·73-s + 0.911·77-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.71592488508334246497303007938, −7.03201068924136096361856720586, −6.42783003885702589004410719258, −5.37778945224415520400717124986, −4.96239295533785831010497332816, −3.97180381159704967733450446023, −3.33182129130337980165299283371, −2.29144961327676055005140790151, −1.34076404227791124225319571078, 0,
1.34076404227791124225319571078, 2.29144961327676055005140790151, 3.33182129130337980165299283371, 3.97180381159704967733450446023, 4.96239295533785831010497332816, 5.37778945224415520400717124986, 6.42783003885702589004410719258, 7.03201068924136096361856720586, 7.71592488508334246497303007938
