| L(s) = 1 | + 4·4-s − 10·9-s + 8·13-s + 8·16-s − 8·17-s + 6·23-s + 19·25-s − 4·27-s − 14·29-s − 40·36-s − 26·43-s − 3·49-s + 32·52-s + 2·53-s + 28·61-s + 12·64-s − 32·68-s + 26·79-s + 45·81-s + 24·92-s + 76·100-s − 16·101-s − 48·103-s − 16·107-s − 16·108-s − 10·113-s − 56·116-s + ⋯ |
| L(s) = 1 | + 2·4-s − 3.33·9-s + 2.21·13-s + 2·16-s − 1.94·17-s + 1.25·23-s + 19/5·25-s − 0.769·27-s − 2.59·29-s − 6.66·36-s − 3.96·43-s − 3/7·49-s + 4.43·52-s + 0.274·53-s + 3.58·61-s + 3/2·64-s − 3.88·68-s + 2.92·79-s + 5·81-s + 2.50·92-s + 38/5·100-s − 1.59·101-s − 4.72·103-s − 1.54·107-s − 1.53·108-s − 0.940·113-s − 5.19·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.065218256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.065218256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( ( 1 + T^{2} )^{3} \) |
| 13 | \( 1 - 8 T + 7 T^{2} + 64 T^{3} + 7 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 2 | \( 1 - p^{2} T^{2} + p^{3} T^{4} - 3 p^{2} T^{6} + p^{5} T^{8} - p^{6} T^{10} + p^{6} T^{12} \) |
| 3 | \( ( 1 + 5 T^{2} + 2 T^{3} + 5 p T^{4} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 - 19 T^{2} + 162 T^{4} - 919 T^{6} + 162 p^{2} T^{8} - 19 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 38 T^{2} + 747 T^{4} - 9800 T^{6} + 747 p^{2} T^{8} - 38 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 4 T + 43 T^{2} + 6 p T^{3} + 43 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 + 5 T^{2} + 238 T^{4} + 10877 T^{6} + 238 p^{2} T^{8} + 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 3 T + 44 T^{2} - 59 T^{3} + 44 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 7 T + 66 T^{2} + 411 T^{3} + 66 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 103 T^{2} + 5914 T^{4} - 224059 T^{6} + 5914 p^{2} T^{8} - 103 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 114 T^{2} + 6171 T^{4} - 242912 T^{6} + 6171 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 138 T^{2} + 10367 T^{4} - 522380 T^{6} + 10367 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 13 T + 164 T^{2} + 1101 T^{3} + 164 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 131 T^{2} + 10566 T^{4} - 603323 T^{6} + 10566 p^{2} T^{8} - 131 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - T + 150 T^{2} - 93 T^{3} + 150 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 286 T^{2} + 36983 T^{4} - 2780916 T^{6} + 36983 p^{2} T^{8} - 286 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 14 T + 211 T^{2} - 1556 T^{3} + 211 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 322 T^{2} + 47303 T^{4} - 4048956 T^{6} + 47303 p^{2} T^{8} - 322 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 122 T^{2} + 17115 T^{4} - 1124048 T^{6} + 17115 p^{2} T^{8} - 122 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 175 T^{2} + 10862 T^{4} - 497775 T^{6} + 10862 p^{2} T^{8} - 175 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 13 T + 200 T^{2} - 1869 T^{3} + 200 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 271 T^{2} + 41066 T^{4} - 4200123 T^{6} + 41066 p^{2} T^{8} - 271 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 415 T^{2} + 80838 T^{4} - 9173143 T^{6} + 80838 p^{2} T^{8} - 415 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 7 T^{2} + 14150 T^{4} + 326769 T^{6} + 14150 p^{2} T^{8} - 7 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.253813798608273811713691242534, −7.926224027525450607697817132418, −7.53539649918199473425592615145, −7.13783554861322453754111723515, −7.11616696485486909006267447342, −6.81074850700636120301994865983, −6.67216906210236000436164792559, −6.56026311914157551545955297033, −6.35649436484857851357481458100, −6.22676806309552780090629189307, −5.68127986712893890694266321547, −5.56018821179401057661646580728, −5.37440888506246759159074062061, −5.33654674742632997477513696031, −4.96348232972901228501914286517, −4.60458169138919007256741610810, −4.18756715711776749617986839205, −3.58782396749804202945136524624, −3.53525046288720474932622155674, −3.34558627209866481755641184788, −2.89522158791828787790384201645, −2.76963395708607092435885909491, −2.27275681099378815718370678345, −2.06041210560032184859029994599, −1.34673579296682142216681795402,
1.34673579296682142216681795402, 2.06041210560032184859029994599, 2.27275681099378815718370678345, 2.76963395708607092435885909491, 2.89522158791828787790384201645, 3.34558627209866481755641184788, 3.53525046288720474932622155674, 3.58782396749804202945136524624, 4.18756715711776749617986839205, 4.60458169138919007256741610810, 4.96348232972901228501914286517, 5.33654674742632997477513696031, 5.37440888506246759159074062061, 5.56018821179401057661646580728, 5.68127986712893890694266321547, 6.22676806309552780090629189307, 6.35649436484857851357481458100, 6.56026311914157551545955297033, 6.67216906210236000436164792559, 6.81074850700636120301994865983, 7.11616696485486909006267447342, 7.13783554861322453754111723515, 7.53539649918199473425592615145, 7.926224027525450607697817132418, 8.253813798608273811713691242534
Plot not available for L-functions of degree greater than 10.
