| L(s) = 1 | + 16-s − 2·25-s + 4·67-s − 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 16-s − 2·25-s + 4·67-s − 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5084325896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5084325896\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−8.385762160058102779869942834926, −7.889531821328608948275817454481, −7.84273445823638068332700752822, −7.59703851389044336707338211636, −7.51647339343988795042853942915, −6.98762824746807386839413737328, −6.73258232742497397747311064116, −6.65004984613259944318897430329, −6.36194329576796973093361987275, −5.99673600071139469592678931300, −5.63783728162440250647776145816, −5.58395740293661999262320915742, −5.41790831006616825823780459578, −4.91076093647297368638248374155, −4.83744785579779234334522478818, −4.24701759009354368989110402291, −4.03905757834259701265667766827, −3.86931961490569081585999189777, −3.53857529263476151221789652631, −3.25394871834383751516281348352, −2.71155808384978503277722180903, −2.47891726023275405362271729690, −2.14017190261486800664524466824, −1.51270748961077959396563896666, −1.22447069934551854485032977313,
1.22447069934551854485032977313, 1.51270748961077959396563896666, 2.14017190261486800664524466824, 2.47891726023275405362271729690, 2.71155808384978503277722180903, 3.25394871834383751516281348352, 3.53857529263476151221789652631, 3.86931961490569081585999189777, 4.03905757834259701265667766827, 4.24701759009354368989110402291, 4.83744785579779234334522478818, 4.91076093647297368638248374155, 5.41790831006616825823780459578, 5.58395740293661999262320915742, 5.63783728162440250647776145816, 5.99673600071139469592678931300, 6.36194329576796973093361987275, 6.65004984613259944318897430329, 6.73258232742497397747311064116, 6.98762824746807386839413737328, 7.51647339343988795042853942915, 7.59703851389044336707338211636, 7.84273445823638068332700752822, 7.889531821328608948275817454481, 8.385762160058102779869942834926
