| L(s) = 1 | + 4-s − 2·7-s − 4·13-s + 4·16-s − 16·19-s − 2·25-s − 2·28-s + 8·31-s + 8·37-s + 8·43-s + 49-s − 4·52-s + 20·61-s + 11·64-s + 8·67-s + 56·73-s − 16·76-s − 16·79-s + 8·91-s − 28·97-s − 2·100-s + 8·103-s + 8·109-s − 8·112-s + 10·121-s + 8·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 1.10·13-s + 16-s − 3.67·19-s − 2/5·25-s − 0.377·28-s + 1.43·31-s + 1.31·37-s + 1.21·43-s + 1/7·49-s − 0.554·52-s + 2.56·61-s + 11/8·64-s + 0.977·67-s + 6.55·73-s − 1.83·76-s − 1.80·79-s + 0.838·91-s − 2.84·97-s − 1/5·100-s + 0.788·103-s + 0.766·109-s − 0.755·112-s + 0.909·121-s + 0.718·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.269726297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.269726297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^3$ | \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 26 T^{2} - 1005 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 46 T^{2} - 93 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 70 T^{2} + 1419 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 166 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \) |
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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
−7.997396278611694818498002768736, −7.46653983252148169539255046945, −7.18035384163905091696878695690, −6.87674200832465847078516757210, −6.79623694807527420097752725568, −6.59529482658329391243123707675, −6.44498463330237847956534700962, −6.04698237815152493125001954417, −5.93280252992874800060660115368, −5.58329743053943267709444836293, −5.29091289216703518474563171252, −5.21590318024642083996024362424, −4.48555863951209342944657356987, −4.41333633923385585149827293939, −4.40664850805581626324290538819, −3.93664318576452541472925165667, −3.67056451189458619865424150178, −3.29083657515027176306926193048, −3.04323684556029606517731640544, −2.35527024725192331897776410508, −2.31227417119174298209987017253, −2.28847172101989537014774372157, −1.79393065669732477356681612123, −0.77910926802181957947642901557, −0.62310919697730656263421114287,
0.62310919697730656263421114287, 0.77910926802181957947642901557, 1.79393065669732477356681612123, 2.28847172101989537014774372157, 2.31227417119174298209987017253, 2.35527024725192331897776410508, 3.04323684556029606517731640544, 3.29083657515027176306926193048, 3.67056451189458619865424150178, 3.93664318576452541472925165667, 4.40664850805581626324290538819, 4.41333633923385585149827293939, 4.48555863951209342944657356987, 5.21590318024642083996024362424, 5.29091289216703518474563171252, 5.58329743053943267709444836293, 5.93280252992874800060660115368, 6.04698237815152493125001954417, 6.44498463330237847956534700962, 6.59529482658329391243123707675, 6.79623694807527420097752725568, 6.87674200832465847078516757210, 7.18035384163905091696878695690, 7.46653983252148169539255046945, 7.997396278611694818498002768736
