| L(s) = 1 | − 3-s − 2·4-s + 2·12-s + 4·16-s + 10·19-s + 2·25-s + 4·27-s + 10·43-s − 4·48-s − 10·49-s − 10·57-s + 6·59-s − 8·64-s + 16·67-s + 4·73-s − 2·75-s − 20·76-s − 7·81-s − 18·83-s + 5·89-s − 8·97-s − 4·100-s − 12·107-s − 8·108-s + 12·113-s − 22·121-s + 127-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.577·12-s + 16-s + 2.29·19-s + 2/5·25-s + 0.769·27-s + 1.52·43-s − 0.577·48-s − 1.42·49-s − 1.32·57-s + 0.781·59-s − 64-s + 1.95·67-s + 0.468·73-s − 0.230·75-s − 2.29·76-s − 7/9·81-s − 1.97·83-s + 0.529·89-s − 0.812·97-s − 2/5·100-s − 1.16·107-s − 0.769·108-s + 1.12·113-s − 2·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7509154955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7509154955\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−11.10464667174994854383445586851, −10.43182765468048811537921244636, −9.755372994276823503924378684140, −9.577111051257869202799069615387, −8.882593216158331032113673411095, −8.282759050933229416454512244655, −7.70472320523039873466913349452, −7.09893349505107676771996226945, −6.37121166258695091289747188413, −5.48391706802058211157726910574, −5.28081200921527097007273653872, −4.50741243820507976591418635128, −3.67273767601823770587702432857, −2.86507869493194173339961887338, −1.06675447319116235390859832178,
1.06675447319116235390859832178, 2.86507869493194173339961887338, 3.67273767601823770587702432857, 4.50741243820507976591418635128, 5.28081200921527097007273653872, 5.48391706802058211157726910574, 6.37121166258695091289747188413, 7.09893349505107676771996226945, 7.70472320523039873466913349452, 8.282759050933229416454512244655, 8.882593216158331032113673411095, 9.577111051257869202799069615387, 9.755372994276823503924378684140, 10.43182765468048811537921244636, 11.10464667174994854383445586851
