| L(s) = 1 | − 2·4-s + 3·16-s + 25-s + 6·31-s + 2·49-s − 2·61-s − 4·64-s − 6·79-s − 2·100-s − 4·109-s − 2·121-s − 12·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s + 25-s + 6·31-s + 2·49-s − 2·61-s − 4·64-s − 6·79-s − 2·100-s − 4·109-s − 2·121-s − 12·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7058714414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7058714414\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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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
−7.02336254890689043658756510105, −6.67641740181480838196053470600, −6.43831203277864237896773909676, −6.11578502090099886394843767376, −6.01801089571756233961904196346, −5.83793345669545772773312487639, −5.68526014918180124435450259358, −5.24645850017407491451162262342, −5.07525055456791805950029516436, −4.95648371716187288054778374371, −4.62437775692355609034279256720, −4.38597471077108214470574992390, −4.35306327982578892167889707513, −4.13788599992158762206052105524, −4.01684760411957592133259578399, −3.53447349436593012891692653401, −3.18234470034622892115425425928, −2.99412046526529264398666233906, −2.78164025556725564333853568695, −2.49858949313619016988106553230, −2.49798537604378542099073949450, −1.43622322828939453511757796115, −1.32091325848621284155959167961, −1.21572576887448141047538343544, −0.60702152034050772066015842305,
0.60702152034050772066015842305, 1.21572576887448141047538343544, 1.32091325848621284155959167961, 1.43622322828939453511757796115, 2.49798537604378542099073949450, 2.49858949313619016988106553230, 2.78164025556725564333853568695, 2.99412046526529264398666233906, 3.18234470034622892115425425928, 3.53447349436593012891692653401, 4.01684760411957592133259578399, 4.13788599992158762206052105524, 4.35306327982578892167889707513, 4.38597471077108214470574992390, 4.62437775692355609034279256720, 4.95648371716187288054778374371, 5.07525055456791805950029516436, 5.24645850017407491451162262342, 5.68526014918180124435450259358, 5.83793345669545772773312487639, 6.01801089571756233961904196346, 6.11578502090099886394843767376, 6.43831203277864237896773909676, 6.67641740181480838196053470600, 7.02336254890689043658756510105
