| L(s) = 1 | + 18·9-s + 80·17-s + 44·29-s − 208·37-s − 68·41-s + 178·49-s − 64·53-s + 100·61-s + 80·73-s + 181·81-s − 76·89-s + 208·97-s − 332·101-s + 220·109-s + 192·113-s + 166·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.44e3·153-s + 157-s + 163-s + 167-s − 598·169-s + ⋯ |
| L(s) = 1 | + 2·9-s + 4.70·17-s + 1.51·29-s − 5.62·37-s − 1.65·41-s + 3.63·49-s − 1.20·53-s + 1.63·61-s + 1.09·73-s + 2.23·81-s − 0.853·89-s + 2.14·97-s − 3.28·101-s + 2.01·109-s + 1.69·113-s + 1.37·121-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 + 9.41·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 3.53·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.978778389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.978778389\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2 p^{2} T^{2} + 143 T^{4} - 1052 T^{6} + 143 p^{4} T^{8} - 2 p^{10} T^{10} + p^{12} T^{12} \) |
| 7 | \( 1 - 178 T^{2} + 14783 T^{4} - 829212 T^{6} + 14783 p^{4} T^{8} - 178 p^{8} T^{10} + p^{12} T^{12} \) |
| 11 | \( 1 - 166 T^{2} + 35359 T^{4} - 4679188 T^{6} + 35359 p^{4} T^{8} - 166 p^{8} T^{10} + p^{12} T^{12} \) |
| 13 | \( ( 1 + 23 p T^{2} + 64 p T^{3} + 23 p^{3} T^{4} + p^{6} T^{6} )^{2} \) |
| 17 | \( ( 1 - 40 T + 1123 T^{2} - 20560 T^{3} + 1123 p^{2} T^{4} - 40 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 19 | \( 1 - 454 T^{2} + 15935 T^{4} + 24240300 T^{6} + 15935 p^{4} T^{8} - 454 p^{8} T^{10} + p^{12} T^{12} \) |
| 23 | \( 1 - 1714 T^{2} + 1754239 T^{4} - 1108856092 T^{6} + 1754239 p^{4} T^{8} - 1714 p^{8} T^{10} + p^{12} T^{12} \) |
| 29 | \( ( 1 - 22 T + 1575 T^{2} - 39124 T^{3} + 1575 p^{2} T^{4} - 22 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 31 | \( 1 - 3206 T^{2} + 5912719 T^{4} - 6798206228 T^{6} + 5912719 p^{4} T^{8} - 3206 p^{8} T^{10} + p^{12} T^{12} \) |
| 37 | \( ( 1 + 104 T + 6811 T^{2} + 305552 T^{3} + 6811 p^{2} T^{4} + 104 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 41 | \( ( 1 + 34 T + 4943 T^{2} + 109308 T^{3} + 4943 p^{2} T^{4} + 34 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 43 | \( 1 - 6834 T^{2} + 22234799 T^{4} - 48099402332 T^{6} + 22234799 p^{4} T^{8} - 6834 p^{8} T^{10} + p^{12} T^{12} \) |
| 47 | \( 1 - 4946 T^{2} + 17102815 T^{4} - 48355312220 T^{6} + 17102815 p^{4} T^{8} - 4946 p^{8} T^{10} + p^{12} T^{12} \) |
| 53 | \( ( 1 + 32 T + 2459 T^{2} - 44544 T^{3} + 2459 p^{2} T^{4} + 32 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 59 | \( 1 - 8422 T^{2} + 50180063 T^{4} - 204193825428 T^{6} + 50180063 p^{4} T^{8} - 8422 p^{8} T^{10} + p^{12} T^{12} \) |
| 61 | \( ( 1 - 50 T + 9431 T^{2} - 290556 T^{3} + 9431 p^{2} T^{4} - 50 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 67 | \( 1 - 12754 T^{2} + 78010639 T^{4} - 366668429212 T^{6} + 78010639 p^{4} T^{8} - 12754 p^{8} T^{10} + p^{12} T^{12} \) |
| 71 | \( 1 - 21862 T^{2} + 231858863 T^{4} - 1468469748948 T^{6} + 231858863 p^{4} T^{8} - 21862 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( ( 1 - 40 T + 14579 T^{2} - 370512 T^{3} + 14579 p^{2} T^{4} - 40 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 79 | \( 1 - 11590 T^{2} + 98296655 T^{4} - 773190011028 T^{6} + 98296655 p^{4} T^{8} - 11590 p^{8} T^{10} + p^{12} T^{12} \) |
| 83 | \( 1 - 17170 T^{2} + 191959631 T^{4} - 1435306239516 T^{6} + 191959631 p^{4} T^{8} - 17170 p^{8} T^{10} + p^{12} T^{12} \) |
| 89 | \( ( 1 + 38 T + 9823 T^{2} + 446996 T^{3} + 9823 p^{2} T^{4} + 38 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 97 | \( ( 1 - 104 T + 29891 T^{2} - 1963728 T^{3} + 29891 p^{2} T^{4} - 104 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.78808661369715559183493594323, −4.58221346203334676211609213618, −4.33526364105787215275758667208, −4.30917890847904323589385499168, −4.11945160222684127568021564255, −3.79309720685053867171627156612, −3.69587531607374404663042618193, −3.47364942917072218120918699489, −3.43741752429091942600233023793, −3.38547029116695125653982763868, −3.35542671149185985272158346344, −2.94269463565828836907870786644, −2.92219315957107224296860387090, −2.56275866562213296391603383820, −2.24211967264344460531857302821, −2.13621862822245247647153813865, −2.01263718265760410359183276820, −1.71015003918737241145043253374, −1.47798321845319371302011107773, −1.37657054810773292176650927807, −1.24915308910152906470942375105, −0.892982140475798709781439264174, −0.77378107335555803389173823422, −0.63930436804989770749245841404, −0.12099100731277966430399837577,
0.12099100731277966430399837577, 0.63930436804989770749245841404, 0.77378107335555803389173823422, 0.892982140475798709781439264174, 1.24915308910152906470942375105, 1.37657054810773292176650927807, 1.47798321845319371302011107773, 1.71015003918737241145043253374, 2.01263718265760410359183276820, 2.13621862822245247647153813865, 2.24211967264344460531857302821, 2.56275866562213296391603383820, 2.92219315957107224296860387090, 2.94269463565828836907870786644, 3.35542671149185985272158346344, 3.38547029116695125653982763868, 3.43741752429091942600233023793, 3.47364942917072218120918699489, 3.69587531607374404663042618193, 3.79309720685053867171627156612, 4.11945160222684127568021564255, 4.30917890847904323589385499168, 4.33526364105787215275758667208, 4.58221346203334676211609213618, 4.78808661369715559183493594323
Plot not available for L-functions of degree greater than 10.
