| L(s) = 1 | − 3-s + 9-s + 8·11-s + 4·17-s − 8·19-s − 16·23-s − 6·25-s − 27-s + 12·29-s + 8·31-s − 8·33-s − 14·49-s − 4·51-s − 4·53-s + 8·57-s − 8·67-s + 16·69-s + 6·75-s + 81-s − 8·83-s − 12·87-s − 12·89-s − 8·93-s + 4·97-s + 8·99-s + 32·103-s − 4·109-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 2.41·11-s + 0.970·17-s − 1.83·19-s − 3.33·23-s − 6/5·25-s − 0.192·27-s + 2.22·29-s + 1.43·31-s − 1.39·33-s − 2·49-s − 0.560·51-s − 0.549·53-s + 1.05·57-s − 0.977·67-s + 1.92·69-s + 0.692·75-s + 1/9·81-s − 0.878·83-s − 1.28·87-s − 1.27·89-s − 0.829·93-s + 0.406·97-s + 0.804·99-s + 3.15·103-s − 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1660608 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1660608 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 31 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.83836664553207816237922609050, −7.01782638700613378711892827179, −6.49036642329105106750290215264, −6.42897107072744719896577758454, −5.96952370785020966524836071168, −5.82635205587879445324667459884, −4.80289467100193947579717711472, −4.33498672996092878289134306443, −4.25303028692796488061253338187, −3.69023092290198626402165503466, −3.12663176891661834294853250910, −2.18779217702540813671625713145, −1.73127362156083975310886909969, −1.11138707404211957515535400621, 0,
1.11138707404211957515535400621, 1.73127362156083975310886909969, 2.18779217702540813671625713145, 3.12663176891661834294853250910, 3.69023092290198626402165503466, 4.25303028692796488061253338187, 4.33498672996092878289134306443, 4.80289467100193947579717711472, 5.82635205587879445324667459884, 5.96952370785020966524836071168, 6.42897107072744719896577758454, 6.49036642329105106750290215264, 7.01782638700613378711892827179, 7.83836664553207816237922609050
