| L(s) = 1 | + 4·3-s − 2·5-s + 2·7-s + 6·9-s + 2·11-s + 2·13-s − 8·15-s + 16·19-s + 8·21-s − 6·23-s + 11·25-s − 4·27-s + 10·29-s + 10·31-s + 8·33-s − 4·35-s − 16·37-s + 8·39-s + 14·41-s − 10·43-s − 12·45-s + 2·47-s + 9·49-s − 16·53-s − 4·55-s + 64·57-s − 14·59-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s + 0.755·7-s + 2·9-s + 0.603·11-s + 0.554·13-s − 2.06·15-s + 3.67·19-s + 1.74·21-s − 1.25·23-s + 11/5·25-s − 0.769·27-s + 1.85·29-s + 1.79·31-s + 1.39·33-s − 0.676·35-s − 2.63·37-s + 1.28·39-s + 2.18·41-s − 1.52·43-s − 1.78·45-s + 0.291·47-s + 9/7·49-s − 2.19·53-s − 0.539·55-s + 8.47·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.526243371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.526243371\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 2 T - 5 T^{2} + 10 T^{3} + 4 T^{4} + 10 p T^{5} - 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 13 T^{2} + 10 T^{3} + 124 T^{4} + 10 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 46 T^{3} - 212 T^{4} + 46 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - 13 T^{2} + 18 T^{3} + 1044 T^{4} + 18 p T^{5} - 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 10 T + 41 T^{2} - 10 T^{3} - 260 T^{4} - 10 p T^{5} + 41 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 10 T + 19 T^{2} - 190 T^{3} + 2500 T^{4} - 190 p T^{5} + 19 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 14 T + 89 T^{2} - 350 T^{3} + 1732 T^{4} - 350 p T^{5} + 89 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 47 | $D_4\times C_2$ | \( 1 - 2 T - 37 T^{2} + 106 T^{3} - 716 T^{4} + 106 p T^{5} - 37 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 14 T + 83 T^{2} - 70 T^{3} - 2276 T^{4} - 70 p T^{5} + 83 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 6 T - 71 T^{2} - 90 T^{3} + 5532 T^{4} - 90 p T^{5} - 71 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 10 T - 5 T^{2} - 290 T^{3} - 164 T^{4} - 290 p T^{5} - 5 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 22 T + 211 T^{2} - 2530 T^{3} + 30052 T^{4} - 2530 p T^{5} + 211 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6 T - 133 T^{2} - 18 T^{3} + 18684 T^{4} - 18 p T^{5} - 133 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 16 T + 218 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 167 T^{2} - 46 T^{3} + 19444 T^{4} - 46 p T^{5} - 167 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.919990534893391139527431022185, −7.57300857532943257530523585071, −7.41224954358226739939056998737, −7.22137311280688918980388961774, −6.84958210846295113531646742289, −6.80747250635659838157061923861, −6.23983221875339386927075768264, −6.09613362390045556365967160179, −5.86224862027021403070764277253, −5.51521547616245808768252320713, −5.14450479600244695432608722570, −4.85814454422634889427170534330, −4.70538099372624309367350085496, −4.36868652569293850968624610200, −4.29475107872026393678415096385, −3.50842229530506274032217200578, −3.41686170466394717851233553985, −3.39059690869868641099343918450, −3.18698482994409146676985043365, −2.79159711250996208435376553078, −2.48120016549726666044315873454, −2.08323582182060101802531385446, −1.46898640719925528769004951341, −1.30174529726152120009415348975, −0.77996947509189724206880637229,
0.77996947509189724206880637229, 1.30174529726152120009415348975, 1.46898640719925528769004951341, 2.08323582182060101802531385446, 2.48120016549726666044315873454, 2.79159711250996208435376553078, 3.18698482994409146676985043365, 3.39059690869868641099343918450, 3.41686170466394717851233553985, 3.50842229530506274032217200578, 4.29475107872026393678415096385, 4.36868652569293850968624610200, 4.70538099372624309367350085496, 4.85814454422634889427170534330, 5.14450479600244695432608722570, 5.51521547616245808768252320713, 5.86224862027021403070764277253, 6.09613362390045556365967160179, 6.23983221875339386927075768264, 6.80747250635659838157061923861, 6.84958210846295113531646742289, 7.22137311280688918980388961774, 7.41224954358226739939056998737, 7.57300857532943257530523585071, 7.919990534893391139527431022185
