| L(s) = 1 | − 3-s − 6·7-s − 2·9-s − 4·16-s + 6·21-s − 6·25-s + 5·27-s − 2·37-s + 4·48-s + 13·49-s + 12·63-s + 24·67-s − 18·73-s + 6·75-s + 81-s + 2·111-s + 24·112-s − 13·121-s + 127-s + 131-s + 137-s + 139-s + 8·144-s − 13·147-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2.26·7-s − 2/3·9-s − 16-s + 1.30·21-s − 6/5·25-s + 0.962·27-s − 0.328·37-s + 0.577·48-s + 13/7·49-s + 1.51·63-s + 2.93·67-s − 2.10·73-s + 0.692·75-s + 1/9·81-s + 0.189·111-s + 2.26·112-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2/3·144-s − 1.07·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−11.16029971999182991696528816390, −10.41061753829729616891739757447, −9.912768392113285810277003943687, −9.478920738330541292036115084853, −8.959966202055794316130571000247, −8.317274067066479231050626054571, −7.43740545613548175720275545846, −6.69574194693160535319989396184, −6.40392984433162091628313383598, −5.81760095975305901783447756023, −5.10467283470590305776146636253, −4.03550440149603943875494198986, −3.31074022592826201162872689060, −2.48318739661211356857141296311, 0,
2.48318739661211356857141296311, 3.31074022592826201162872689060, 4.03550440149603943875494198986, 5.10467283470590305776146636253, 5.81760095975305901783447756023, 6.40392984433162091628313383598, 6.69574194693160535319989396184, 7.43740545613548175720275545846, 8.317274067066479231050626054571, 8.959966202055794316130571000247, 9.478920738330541292036115084853, 9.912768392113285810277003943687, 10.41061753829729616891739757447, 11.16029971999182991696528816390
