| L(s) = 1 | + 2·5-s − 2·7-s + 4·13-s − 6·17-s − 8·19-s − 2·23-s + 3·25-s + 6·29-s − 14·31-s − 4·35-s − 2·37-s − 6·41-s − 8·43-s − 5·49-s − 6·53-s − 6·59-s + 4·61-s + 8·65-s − 2·67-s − 6·71-s + 4·73-s − 8·79-s + 6·83-s − 12·85-s + 24·89-s − 8·91-s − 16·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s + 1.10·13-s − 1.45·17-s − 1.83·19-s − 0.417·23-s + 3/5·25-s + 1.11·29-s − 2.51·31-s − 0.676·35-s − 0.328·37-s − 0.937·41-s − 1.21·43-s − 5/7·49-s − 0.824·53-s − 0.781·59-s + 0.512·61-s + 0.992·65-s − 0.244·67-s − 0.712·71-s + 0.468·73-s − 0.900·79-s + 0.658·83-s − 1.30·85-s + 2.54·89-s − 0.838·91-s − 1.64·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17139600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17139600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 61 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 129 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 96 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 121 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20 T + 270 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.215062155958794478784712367833, −8.184144373530791776156606761460, −7.33403343027783695082205517342, −7.05422398999643817662684335237, −6.55282847677968303761364243120, −6.46675581517503817474233888395, −6.12401257948697757839470333869, −5.84778219585465545404250680930, −5.14539550830265368180116669930, −4.98642706046647012023421693489, −4.44427505255763618594111199456, −3.99693223423124269570719343192, −3.54387774498123272103179296335, −3.31944114817015098863219905235, −2.45978184089193883410605721198, −2.34908079741243186956872930816, −1.58364564112324276995706147920, −1.45222310834880214385841446354, 0, 0,
1.45222310834880214385841446354, 1.58364564112324276995706147920, 2.34908079741243186956872930816, 2.45978184089193883410605721198, 3.31944114817015098863219905235, 3.54387774498123272103179296335, 3.99693223423124269570719343192, 4.44427505255763618594111199456, 4.98642706046647012023421693489, 5.14539550830265368180116669930, 5.84778219585465545404250680930, 6.12401257948697757839470333869, 6.46675581517503817474233888395, 6.55282847677968303761364243120, 7.05422398999643817662684335237, 7.33403343027783695082205517342, 8.184144373530791776156606761460, 8.215062155958794478784712367833
