| L(s) = 1 | + 3·5-s + 7-s − 4·11-s + 3·13-s + 4·17-s + 23-s + 4·25-s + 3·29-s + 6·31-s + 3·35-s + 9·37-s − 9·41-s − 3·43-s − 7·47-s + 49-s − 4·53-s − 12·55-s − 6·59-s − 10·61-s + 9·65-s + 4·67-s − 6·71-s − 8·73-s − 4·77-s − 8·79-s − 4·83-s + 12·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s − 1.20·11-s + 0.832·13-s + 0.970·17-s + 0.208·23-s + 4/5·25-s + 0.557·29-s + 1.07·31-s + 0.507·35-s + 1.47·37-s − 1.40·41-s − 0.457·43-s − 1.02·47-s + 1/7·49-s − 0.549·53-s − 1.61·55-s − 0.781·59-s − 1.28·61-s + 1.11·65-s + 0.488·67-s − 0.712·71-s − 0.936·73-s − 0.455·77-s − 0.900·79-s − 0.439·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.92018467857116, −13.47919791210261, −13.31360477620658, −12.78589752436097, −12.05383244789968, −11.72483198915308, −10.89909681941277, −10.63363497237948, −10.08096788375674, −9.705064191150680, −9.246967288337048, −8.381096240384981, −8.211291340403477, −7.641655306719905, −6.892747678135333, −6.296218138044984, −5.899137064515182, −5.407765052935314, −4.856599148543706, −4.393202744927612, −3.366683449564583, −2.906013869760671, −2.365983083450185, −1.486674347893377, −1.203094841913124, 0,
1.203094841913124, 1.486674347893377, 2.365983083450185, 2.906013869760671, 3.366683449564583, 4.393202744927612, 4.856599148543706, 5.407765052935314, 5.899137064515182, 6.296218138044984, 6.892747678135333, 7.641655306719905, 8.211291340403477, 8.381096240384981, 9.246967288337048, 9.705064191150680, 10.08096788375674, 10.63363497237948, 10.89909681941277, 11.72483198915308, 12.05383244789968, 12.78589752436097, 13.31360477620658, 13.47919791210261, 13.92018467857116
