| L(s) = 1 | + 2·3-s + 2·5-s + 4·11-s − 2·13-s + 4·15-s + 8·17-s + 2·19-s − 4·23-s + 3·25-s − 2·27-s − 8·31-s + 8·33-s + 2·37-s − 4·39-s + 8·41-s + 8·43-s − 8·47-s − 14·49-s + 16·51-s + 6·53-s + 8·55-s + 4·57-s − 4·59-s + 20·61-s − 4·65-s + 2·67-s − 8·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 1.20·11-s − 0.554·13-s + 1.03·15-s + 1.94·17-s + 0.458·19-s − 0.834·23-s + 3/5·25-s − 0.384·27-s − 1.43·31-s + 1.39·33-s + 0.328·37-s − 0.640·39-s + 1.24·41-s + 1.21·43-s − 1.16·47-s − 2·49-s + 2.24·51-s + 0.824·53-s + 1.07·55-s + 0.529·57-s − 0.520·59-s + 2.56·61-s − 0.496·65-s + 0.244·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.700328783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.700328783\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 108 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 158 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 168 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.212285781996372570485336205319, −7.980144446491688092225540348317, −7.42099350658955456862918128468, −7.41924495635056545109958279238, −6.94525599948907435367806661408, −6.41784693859316038809411745802, −6.00063991177814870627194144538, −5.90504680479009221470345634213, −5.31301760543345704457223461316, −5.22025320791871662652080369632, −4.53536398989479246974153272860, −4.16027388992047232344216832841, −3.65336821952731249485516932161, −3.41855340748133833284406859171, −2.93880545538806908096486447034, −2.69318672665590305609660576736, −1.90320092388514352667133749849, −1.89400778345827285742060859547, −1.16450084645296707077694905432, −0.62902143134760742513834150469,
0.62902143134760742513834150469, 1.16450084645296707077694905432, 1.89400778345827285742060859547, 1.90320092388514352667133749849, 2.69318672665590305609660576736, 2.93880545538806908096486447034, 3.41855340748133833284406859171, 3.65336821952731249485516932161, 4.16027388992047232344216832841, 4.53536398989479246974153272860, 5.22025320791871662652080369632, 5.31301760543345704457223461316, 5.90504680479009221470345634213, 6.00063991177814870627194144538, 6.41784693859316038809411745802, 6.94525599948907435367806661408, 7.41924495635056545109958279238, 7.42099350658955456862918128468, 7.980144446491688092225540348317, 8.212285781996372570485336205319
