| L(s) = 1 | + 2·3-s − 3·5-s − 2·7-s + 3·9-s − 7·13-s − 6·15-s − 2·17-s + 12·19-s − 4·21-s − 2·23-s − 2·25-s + 4·27-s + 8·29-s + 10·31-s + 6·35-s − 10·37-s − 14·39-s + 4·41-s + 43-s − 9·45-s − 6·47-s + 3·49-s − 4·51-s − 5·53-s + 24·57-s + 3·59-s − 7·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s − 0.755·7-s + 9-s − 1.94·13-s − 1.54·15-s − 0.485·17-s + 2.75·19-s − 0.872·21-s − 0.417·23-s − 2/5·25-s + 0.769·27-s + 1.48·29-s + 1.79·31-s + 1.01·35-s − 1.64·37-s − 2.24·39-s + 0.624·41-s + 0.152·43-s − 1.34·45-s − 0.875·47-s + 3/7·49-s − 0.560·51-s − 0.686·53-s + 3.17·57-s + 0.390·59-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 27 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 12 T + 69 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + p T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + p T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 25 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 111 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 109 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 103 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 135 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 21 T + 251 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 127 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 87 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 87 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 17 T + 189 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 105 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.56195718143062792100626603696, −7.52412292762567520064767913132, −7.04755057467137186395322827729, −6.99497442707161891963025716990, −6.38731197934009362436816804483, −6.08917510380939203564963365080, −5.47565733543702613294703081691, −5.09228631331746180848335262872, −4.73407376741707050118322258454, −4.53851231323841241013272825803, −3.92451885214682736216044604619, −3.73327328709512707984184001001, −3.22594583372798767418483121843, −2.97386399381430501815190676565, −2.46745426239782070216495055304, −2.45621736201748562606392639918, −1.32580720577385318735724055540, −1.19895407086349101317427543250, 0, 0,
1.19895407086349101317427543250, 1.32580720577385318735724055540, 2.45621736201748562606392639918, 2.46745426239782070216495055304, 2.97386399381430501815190676565, 3.22594583372798767418483121843, 3.73327328709512707984184001001, 3.92451885214682736216044604619, 4.53851231323841241013272825803, 4.73407376741707050118322258454, 5.09228631331746180848335262872, 5.47565733543702613294703081691, 6.08917510380939203564963365080, 6.38731197934009362436816804483, 6.99497442707161891963025716990, 7.04755057467137186395322827729, 7.52412292762567520064767913132, 7.56195718143062792100626603696
