| L(s) = 1 | − 6·3-s − 3·4-s + 21·9-s + 18·12-s + 5·16-s + 14·23-s − 10·25-s − 54·27-s + 8·31-s − 63·36-s + 10·37-s − 4·47-s − 30·48-s + 2·49-s − 2·53-s − 4·59-s − 3·64-s − 24·67-s − 84·69-s + 2·71-s + 60·75-s + 108·81-s − 28·89-s − 42·92-s − 48·93-s + 2·97-s + 30·100-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 3/2·4-s + 7·9-s + 5.19·12-s + 5/4·16-s + 2.91·23-s − 2·25-s − 10.3·27-s + 1.43·31-s − 10.5·36-s + 1.64·37-s − 0.583·47-s − 4.33·48-s + 2/7·49-s − 0.274·53-s − 0.520·59-s − 3/8·64-s − 2.93·67-s − 10.1·69-s + 0.237·71-s + 6.92·75-s + 12·81-s − 2.96·89-s − 4.37·92-s − 4.97·93-s + 0.203·97-s + 3·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 339889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 339889 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_2$ | \( 1 + p T^{2} \) |
| 53 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−8.564776268282943764220707070932, −7.966923727015514001224791192936, −7.20736947999270954805514319970, −7.12035687614008033346324492241, −6.28372834263201669966540466161, −6.04347889941024738587947019148, −5.66393024096249419924903950670, −5.12910550394150270879712944112, −4.71629718922730321854403792085, −4.50862835068296220400083952922, −4.02042745637930707154786801317, −2.99582779445928276782980514237, −1.38439944793772811124702445306, −0.819378028185293560553368516802, 0,
0.819378028185293560553368516802, 1.38439944793772811124702445306, 2.99582779445928276782980514237, 4.02042745637930707154786801317, 4.50862835068296220400083952922, 4.71629718922730321854403792085, 5.12910550394150270879712944112, 5.66393024096249419924903950670, 6.04347889941024738587947019148, 6.28372834263201669966540466161, 7.12035687614008033346324492241, 7.20736947999270954805514319970, 7.966923727015514001224791192936, 8.564776268282943764220707070932
