| L(s) = 1 | − 6·3-s + 16·9-s − 4·13-s − 6·17-s − 6·19-s − 6·23-s − 14·25-s − 24·27-s − 12·29-s + 8·37-s + 24·39-s − 18·41-s − 36·43-s − 10·49-s + 36·51-s + 36·57-s − 12·59-s − 36·61-s − 6·67-s + 36·69-s + 30·71-s + 84·75-s − 24·79-s + 21·81-s − 24·83-s + 72·87-s − 12·89-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 16/3·9-s − 1.10·13-s − 1.45·17-s − 1.37·19-s − 1.25·23-s − 2.79·25-s − 4.61·27-s − 2.22·29-s + 1.31·37-s + 3.84·39-s − 2.81·41-s − 5.48·43-s − 1.42·49-s + 5.04·51-s + 4.76·57-s − 1.56·59-s − 4.60·61-s − 0.733·67-s + 4.33·69-s + 3.56·71-s + 9.69·75-s − 2.70·79-s + 7/3·81-s − 2.63·83-s + 7.71·87-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 p T + 20 T^{2} + 16 p T^{3} + 91 T^{4} + 16 p^{2} T^{5} + 20 p^{2} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 5 T^{2} - 18 T^{3} + 60 T^{4} - 18 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T - 8 T^{2} + 36 T^{3} + 891 T^{4} + 36 p T^{5} - 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 36 p T^{5} - 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 12 T + 109 T^{2} + 732 T^{3} + 4272 T^{4} + 732 p T^{5} + 109 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 20 T^{2} - 678 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 8 T + T^{2} + 88 T^{3} + 232 T^{4} + 88 p T^{5} + p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 36 T + 622 T^{2} + 6840 T^{3} + 52827 T^{4} + 6840 p T^{5} + 622 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 1818 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 26 T^{2} + 1899 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 38 T^{2} - 144 T^{3} - 741 T^{4} - 144 p T^{5} + 38 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 36 T + 653 T^{2} + 7956 T^{3} + 71472 T^{4} + 7956 p T^{5} + 653 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 6 T - 80 T^{2} - 108 T^{3} + 6555 T^{4} - 108 p T^{5} - 80 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 30 T + 508 T^{2} - 6240 T^{3} + 59523 T^{4} - 6240 p T^{5} + 508 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 94 T^{2} + 552 T^{3} - 1533 T^{4} + 552 p T^{5} + 94 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.72408930626670411408262597742, −7.51688178351082182344109482824, −7.13950416034596584885515816296, −6.83974332914137070798219456170, −6.80090567775694803637690421120, −6.47715012416389482423931812368, −6.43255493432673088387957864134, −6.01685940753180579280486418089, −6.00507102378909348348141838790, −5.72123882800760642411037093895, −5.53510546586307052289615881679, −5.37965047500399563365257126725, −5.09977681012943747586778877994, −4.66932085531096349826805844152, −4.52477829635618333634785034522, −4.42230817266854609772076589345, −4.40493373556656307747180293251, −3.68258941768307417564459961712, −3.42106742100160321591840129485, −3.28261446665017997578343359922, −2.86595080290625693495463666517, −2.03308556137160127111835272593, −1.95926020103333398719218504371, −1.73638610231471740308443801922, −1.55127966613772653635832194470, 0, 0, 0, 0,
1.55127966613772653635832194470, 1.73638610231471740308443801922, 1.95926020103333398719218504371, 2.03308556137160127111835272593, 2.86595080290625693495463666517, 3.28261446665017997578343359922, 3.42106742100160321591840129485, 3.68258941768307417564459961712, 4.40493373556656307747180293251, 4.42230817266854609772076589345, 4.52477829635618333634785034522, 4.66932085531096349826805844152, 5.09977681012943747586778877994, 5.37965047500399563365257126725, 5.53510546586307052289615881679, 5.72123882800760642411037093895, 6.00507102378909348348141838790, 6.01685940753180579280486418089, 6.43255493432673088387957864134, 6.47715012416389482423931812368, 6.80090567775694803637690421120, 6.83974332914137070798219456170, 7.13950416034596584885515816296, 7.51688178351082182344109482824, 7.72408930626670411408262597742
