| L(s) = 1 | − 4·13-s − 16-s + 4·31-s − 4·43-s + 2·49-s + 4·67-s + 4·97-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 4·13-s − 16-s + 4·31-s − 4·43-s + 2·49-s + 4·67-s + 4·97-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3848645773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3848645773\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
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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
−8.353983810850964368888156806829, −8.122050644798931139573399530374, −7.73768074330630743817850565657, −7.58874656069789216065624730099, −7.36313114954289662332727941628, −6.95329677180448468545439805557, −6.90576961626624250002864950432, −6.74305470865466984333616645250, −6.34078008816720197084693177101, −6.17306625928658912296584275359, −5.94129394461114701075032964697, −5.20212322128511737537263313873, −5.07640945539313864995736517812, −4.95121193538855816691733734501, −4.85370867393979454556579685243, −4.71388316553114415104045157819, −4.14131322971884695454873142521, −3.78348720928138463962681262861, −3.64572877319623334347323177999, −2.80006230187750664070888616745, −2.79685177606582793573701173399, −2.59097517581282865672808478937, −2.11170076498663153757087653620, −1.92571696198638370433387376998, −0.906237298384295704270118633130,
0.906237298384295704270118633130, 1.92571696198638370433387376998, 2.11170076498663153757087653620, 2.59097517581282865672808478937, 2.79685177606582793573701173399, 2.80006230187750664070888616745, 3.64572877319623334347323177999, 3.78348720928138463962681262861, 4.14131322971884695454873142521, 4.71388316553114415104045157819, 4.85370867393979454556579685243, 4.95121193538855816691733734501, 5.07640945539313864995736517812, 5.20212322128511737537263313873, 5.94129394461114701075032964697, 6.17306625928658912296584275359, 6.34078008816720197084693177101, 6.74305470865466984333616645250, 6.90576961626624250002864950432, 6.95329677180448468545439805557, 7.36313114954289662332727941628, 7.58874656069789216065624730099, 7.73768074330630743817850565657, 8.122050644798931139573399530374, 8.353983810850964368888156806829
