| L(s) = 1 | + 12·7-s + 12·13-s − 16-s − 12·19-s + 20·37-s + 72·49-s + 56·61-s − 16·67-s − 24·73-s + 32·79-s + 144·91-s − 8·97-s + 32·109-s − 12·112-s + 127-s + 131-s − 144·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 4.53·7-s + 3.32·13-s − 1/4·16-s − 2.75·19-s + 3.28·37-s + 72/7·49-s + 7.17·61-s − 1.95·67-s − 2.80·73-s + 3.60·79-s + 15.0·91-s − 0.812·97-s + 3.06·109-s − 1.13·112-s + 0.0887·127-s + 0.0873·131-s − 12.4·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.84618972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.84618972\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 82 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )( 1 + 80 T^{2} + p^{2} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 878 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 + 5794 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 8722 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 + 4322 T^{4} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.96078746150590906847845175679, −6.75547182626354148992191377431, −6.50350230142202985960548300242, −6.33998401129741245514290231125, −6.06864259110681443535437116304, −5.70677429177499261316670551563, −5.69817195081631741055954291992, −5.52255770282708963836942964000, −5.12291918079759366910969890862, −4.95264512403783795320248053593, −4.49238107020619914212398158944, −4.42224822078629083901649012009, −4.40905802680895216894295404857, −4.05972686581300406401013812438, −4.01564294070419277635390001737, −3.46642646008681120266484724706, −3.39267608574935873607806350406, −2.79272014511095049005951347675, −2.23907821282716359287864862543, −2.16293662999140330359819351078, −1.99311189771777589971081480796, −1.81260174399166227427106875239, −1.15300005651719481996054202051, −0.962965994178585733593886074558, −0.916457705055867508400771129931,
0.916457705055867508400771129931, 0.962965994178585733593886074558, 1.15300005651719481996054202051, 1.81260174399166227427106875239, 1.99311189771777589971081480796, 2.16293662999140330359819351078, 2.23907821282716359287864862543, 2.79272014511095049005951347675, 3.39267608574935873607806350406, 3.46642646008681120266484724706, 4.01564294070419277635390001737, 4.05972686581300406401013812438, 4.40905802680895216894295404857, 4.42224822078629083901649012009, 4.49238107020619914212398158944, 4.95264512403783795320248053593, 5.12291918079759366910969890862, 5.52255770282708963836942964000, 5.69817195081631741055954291992, 5.70677429177499261316670551563, 6.06864259110681443535437116304, 6.33998401129741245514290231125, 6.50350230142202985960548300242, 6.75547182626354148992191377431, 6.96078746150590906847845175679
