| L(s) = 1 | + 4·13-s − 16-s + 4·43-s − 4·61-s − 8·67-s + 4·79-s + 4·97-s + 4·103-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 4·208-s + ⋯ |
| L(s) = 1 | + 4·13-s − 16-s + 4·43-s − 4·61-s − 8·67-s + 4·79-s + 4·97-s + 4·103-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 4·208-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.866010395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.866010395\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 67 | $C_1$ | \( ( 1 + T )^{8} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
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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
−6.34395911253079863992755588639, −6.28804878744290457656649636560, −6.18977513684971683335564709606, −5.92128939355772233324332526796, −5.87122011656231905380225966440, −5.87005217889429243749792127137, −5.86014901763654521353025591785, −4.87647789558870307351524115810, −4.80200654485976270368628064404, −4.75940237151608450439752727147, −4.67505154122390280143155279133, −4.31460274816729762327262066864, −4.02217649545060334502972794743, −3.78030257639767920730206854393, −3.66466909963963894600455414189, −3.22748587578898096533824690291, −3.22721566942045914690209991482, −3.02859785345208158434072111812, −2.69936718627400046150890702690, −2.05799946024511865788737037551, −2.03708726662288201654305360948, −1.84698139170589525640033417564, −1.24981379391915894066428144000, −1.11715492894546494497931521665, −0.78519822039765553627000189864,
0.78519822039765553627000189864, 1.11715492894546494497931521665, 1.24981379391915894066428144000, 1.84698139170589525640033417564, 2.03708726662288201654305360948, 2.05799946024511865788737037551, 2.69936718627400046150890702690, 3.02859785345208158434072111812, 3.22721566942045914690209991482, 3.22748587578898096533824690291, 3.66466909963963894600455414189, 3.78030257639767920730206854393, 4.02217649545060334502972794743, 4.31460274816729762327262066864, 4.67505154122390280143155279133, 4.75940237151608450439752727147, 4.80200654485976270368628064404, 4.87647789558870307351524115810, 5.86014901763654521353025591785, 5.87005217889429243749792127137, 5.87122011656231905380225966440, 5.92128939355772233324332526796, 6.18977513684971683335564709606, 6.28804878744290457656649636560, 6.34395911253079863992755588639
