| L(s) = 1 | − 16·7-s − 16·25-s + 16·31-s − 248·49-s + 224·73-s + 304·79-s − 704·97-s − 688·103-s + 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 744·169-s + 173-s + 256·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2.28·7-s − 0.639·25-s + 0.516·31-s − 5.06·49-s + 3.06·73-s + 3.84·79-s − 7.25·97-s − 6.67·103-s + 0.198·121-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 + 4.40·169-s + 0.00578·173-s + 1.46·175-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.555602474\times10^{-7}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.555602474\times10^{-7}\) |
| \(L(2)\) |
|
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 | \( ( 1 + 8 T^{2} + 914 T^{4} + 8 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 + 2 T + p^{2} T^{2} )^{8} \) |
| 11 | \( ( 1 - 12 T^{2} - 21370 T^{4} - 12 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 372 T^{2} + 69190 T^{4} - 372 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 800 T^{2} + 325634 T^{4} - 800 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 612 T^{2} + 264166 T^{4} - 612 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 564 T^{2} + 436454 T^{4} - 564 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 2600 T^{2} + 3045074 T^{4} + 2600 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 4 T + 518 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2062 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 5888 T^{2} + 14148290 T^{4} - 5888 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 3012 T^{2} + 4690150 T^{4} - 3012 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 1652 T^{2} + 2309030 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 8680 T^{2} + 33219474 T^{4} + 8680 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 1554 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 10268 T^{2} + 48980838 T^{4} - 10268 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 3140 T^{2} + 32832294 T^{4} - 3140 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 12596 T^{2} + 79584614 T^{4} - 12596 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 56 T - 1230 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 76 T + 12518 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + 24244 T^{2} + 241403334 T^{4} + 24244 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 10176 T^{2} + 39402434 T^{4} - 10176 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 176 T + 26210 T^{2} + 176 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−3.55284657131560844711127827345, −3.40087994732330321390355568812, −3.29902685830682125157628359516, −3.28181929126671357094311500046, −3.09460635076697486184917344195, −3.05207760059113523280406623091, −2.70767931911331298430092427524, −2.62245126735371581964928891605, −2.60681576753113305247673364300, −2.59145013733823928328275890950, −2.37005035229868301278338074847, −2.21040701202566463950922909527, −2.16144697075418172191209092738, −1.87033149664828290444242965585, −1.69911003736281196070970296017, −1.47289259880348247993379280572, −1.46676758524875995021299727077, −1.29110372055988459744653970011, −1.15224015869606057604241892388, −1.13743897499460131022825237125, −0.871844282124750169091813733854, −0.44964930339365080587022211099, −0.37409109466220892079051889817, −0.008220077852859006086815901028, −0.00327928874587874504303791337,
0.00327928874587874504303791337, 0.008220077852859006086815901028, 0.37409109466220892079051889817, 0.44964930339365080587022211099, 0.871844282124750169091813733854, 1.13743897499460131022825237125, 1.15224015869606057604241892388, 1.29110372055988459744653970011, 1.46676758524875995021299727077, 1.47289259880348247993379280572, 1.69911003736281196070970296017, 1.87033149664828290444242965585, 2.16144697075418172191209092738, 2.21040701202566463950922909527, 2.37005035229868301278338074847, 2.59145013733823928328275890950, 2.60681576753113305247673364300, 2.62245126735371581964928891605, 2.70767931911331298430092427524, 3.05207760059113523280406623091, 3.09460635076697486184917344195, 3.28181929126671357094311500046, 3.29902685830682125157628359516, 3.40087994732330321390355568812, 3.55284657131560844711127827345
Plot not available for L-functions of degree greater than 10.
