| L(s) = 1 | − 16·7-s − 144·25-s − 240·31-s + 136·49-s − 544·73-s − 976·79-s + 320·97-s + 80·103-s − 744·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 232·169-s + 173-s + 2.30e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2.28·7-s − 5.75·25-s − 7.74·31-s + 2.77·49-s − 7.45·73-s − 12.3·79-s + 3.29·97-s + 0.776·103-s − 6.14·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.37·169-s + 0.00578·173-s + 13.1·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0005592454170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0005592454170\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 72 T^{2} + 98 p^{2} T^{4} + 72 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 + 4 T + 6 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 372 T^{2} + 62342 T^{4} + 372 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 116 T^{2} + 5190 T^{4} - 116 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 288 T^{2} + 184322 T^{4} - 288 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 1124 T^{2} + 551910 T^{4} - 1124 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 858 T^{2} + p^{4} T^{4} )^{4} \) |
| 29 | \( ( 1 + 2152 T^{2} + 2530002 T^{4} + 2152 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 + 60 T + 2726 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 3740 T^{2} + 6501798 T^{4} - 3740 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 2560 T^{2} + 3056322 T^{4} - 2560 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 1732 T^{2} - 375066 T^{4} - 1732 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 6900 T^{2} + 21047462 T^{4} - 6900 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 7720 T^{2} + 28930962 T^{4} + 7720 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 1500 T^{2} + 2678822 T^{4} - 1500 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 6812 T^{2} + 23932518 T^{4} - 6812 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 16196 T^{2} + 105578790 T^{4} - 16196 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 5044 T^{2} + 16873062 T^{4} - 5044 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 136 T + 14898 T^{2} + 136 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 + 244 T + 26502 T^{2} + 244 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 - 1484 T^{2} - 19009338 T^{4} - 1484 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 11712 T^{2} + 138025922 T^{4} - 11712 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 80 T + 4194 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.58080839033863580982945339495, −3.50431835911368687507204001865, −3.29912241431356359440585061080, −3.25236999563271432312928163855, −3.22769802460301738053473178148, −2.96169336834097354255313136891, −2.72560067610033576094344199780, −2.63020337262996548765053383897, −2.58463627242419835605285113592, −2.44425260963381959757564945281, −2.41526244783619506613706232542, −2.33372939417471866803720437339, −1.91970022867571714731343811768, −1.74804675717276220008301483279, −1.73740621299698850228428281813, −1.46335246855625257905858826657, −1.45604659454307153780950902788, −1.41908062309286305306327904220, −1.36758239930099737452851645717, −1.35124651707572352660078272690, −0.55076975769308746395098553299, −0.43898281667443287308237988768, −0.10470774765350301249188376254, −0.06107455031956434836682323431, −0.04832595962199104880216347721,
0.04832595962199104880216347721, 0.06107455031956434836682323431, 0.10470774765350301249188376254, 0.43898281667443287308237988768, 0.55076975769308746395098553299, 1.35124651707572352660078272690, 1.36758239930099737452851645717, 1.41908062309286305306327904220, 1.45604659454307153780950902788, 1.46335246855625257905858826657, 1.73740621299698850228428281813, 1.74804675717276220008301483279, 1.91970022867571714731343811768, 2.33372939417471866803720437339, 2.41526244783619506613706232542, 2.44425260963381959757564945281, 2.58463627242419835605285113592, 2.63020337262996548765053383897, 2.72560067610033576094344199780, 2.96169336834097354255313136891, 3.22769802460301738053473178148, 3.25236999563271432312928163855, 3.29912241431356359440585061080, 3.50431835911368687507204001865, 3.58080839033863580982945339495
Plot not available for L-functions of degree greater than 10.
