| L(s) = 1 | + 2·3-s + 3·4-s − 4·5-s + 9-s + 11-s + 6·12-s − 8·15-s + 5·16-s − 12·20-s + 4·23-s + 2·25-s − 4·27-s + 2·31-s + 2·33-s + 3·36-s − 12·37-s + 3·44-s − 4·45-s − 2·47-s + 10·48-s + 8·49-s − 6·53-s − 4·55-s − 8·59-s − 24·60-s + 3·64-s + 10·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 3/2·4-s − 1.78·5-s + 1/3·9-s + 0.301·11-s + 1.73·12-s − 2.06·15-s + 5/4·16-s − 2.68·20-s + 0.834·23-s + 2/5·25-s − 0.769·27-s + 0.359·31-s + 0.348·33-s + 1/2·36-s − 1.97·37-s + 0.452·44-s − 0.596·45-s − 0.291·47-s + 1.44·48-s + 8/7·49-s − 0.824·53-s − 0.539·55-s − 1.04·59-s − 3.09·60-s + 3/8·64-s + 1.22·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11979 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11979 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.478974579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.478974579\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−11.45131824258061458916450519696, −10.91690577990793610660900803282, −10.45579235275946954710273984552, −9.582086136330591251915885065607, −8.977148289736582097082004206859, −8.275820978254645125631912579542, −7.969960679393019470039862150512, −7.35870763525177012678764393491, −6.99849377937138192702490611689, −6.28826950708592892337652477350, −5.30030269977841569591804308880, −4.19875107801205039988739345512, −3.52202012479345836653568205274, −3.00153015912170044874275189793, −1.95180531308691071472655104039,
1.95180531308691071472655104039, 3.00153015912170044874275189793, 3.52202012479345836653568205274, 4.19875107801205039988739345512, 5.30030269977841569591804308880, 6.28826950708592892337652477350, 6.99849377937138192702490611689, 7.35870763525177012678764393491, 7.969960679393019470039862150512, 8.275820978254645125631912579542, 8.977148289736582097082004206859, 9.582086136330591251915885065607, 10.45579235275946954710273984552, 10.91690577990793610660900803282, 11.45131824258061458916450519696
