| L(s) = 1 | + 3-s − 4·5-s + 9-s − 4·15-s + 8·19-s + 6·25-s + 27-s + 12·29-s + 8·43-s − 4·45-s − 16·47-s − 2·49-s − 4·53-s + 8·57-s − 8·67-s − 4·73-s + 6·75-s + 81-s + 12·87-s − 32·95-s + 4·97-s + 12·101-s + 10·121-s − 4·125-s + 127-s + 8·129-s + 131-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.03·15-s + 1.83·19-s + 6/5·25-s + 0.192·27-s + 2.22·29-s + 1.21·43-s − 0.596·45-s − 2.33·47-s − 2/7·49-s − 0.549·53-s + 1.05·57-s − 0.977·67-s − 0.468·73-s + 0.692·75-s + 1/9·81-s + 1.28·87-s − 3.28·95-s + 0.406·97-s + 1.19·101-s + 0.909·121-s − 0.357·125-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.547810552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.547810552\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 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$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 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
−8.362397925528162486912706424384, −8.188075546752218895922281167497, −7.66231774966848140393613646536, −7.43600379622525299720199607813, −6.93104561640268633693070834164, −6.39912465187101425808576817914, −5.83628968043674094798122319748, −5.00578090596163579524507130472, −4.71421125922396040170294949564, −4.19983851833618222536474458010, −3.57134243093003498063723970037, −3.14260113697421981328547222606, −2.76012234901913998810403748883, −1.59040640156258629858061137421, −0.68549613931472928642814089977,
0.68549613931472928642814089977, 1.59040640156258629858061137421, 2.76012234901913998810403748883, 3.14260113697421981328547222606, 3.57134243093003498063723970037, 4.19983851833618222536474458010, 4.71421125922396040170294949564, 5.00578090596163579524507130472, 5.83628968043674094798122319748, 6.39912465187101425808576817914, 6.93104561640268633693070834164, 7.43600379622525299720199607813, 7.66231774966848140393613646536, 8.188075546752218895922281167497, 8.362397925528162486912706424384
