| L(s) = 1 | − 5·4-s + 4·5-s + 12·9-s + 16-s − 20·20-s + 24·25-s − 184·29-s − 60·36-s − 256·41-s + 48·45-s − 208·49-s + 304·61-s + 35·64-s + 4·80-s + 90·81-s + 560·89-s − 120·100-s + 296·101-s − 608·109-s + 920·116-s + 272·121-s + 204·125-s + 127-s + 131-s + 137-s + 139-s + 12·144-s + ⋯ |
| L(s) = 1 | − 5/4·4-s + 4/5·5-s + 4/3·9-s + 1/16·16-s − 20-s + 0.959·25-s − 6.34·29-s − 5/3·36-s − 6.24·41-s + 1.06·45-s − 4.24·49-s + 4.98·61-s + 0.546·64-s + 1/20·80-s + 10/9·81-s + 6.29·89-s − 6/5·100-s + 2.93·101-s − 5.57·109-s + 7.93·116-s + 2.24·121-s + 1.63·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1/12·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8053313507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8053313507\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5 T^{2} + 3 p^{3} T^{4} + 5 p^{4} T^{6} + p^{8} T^{8} \) |
| 3 | \( ( 1 - p T^{2} )^{4} \) |
| 5 | \( ( 1 - 2 T - 6 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| good | 7 | \( ( 1 + 104 T^{2} + 5454 T^{4} + 104 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 136 T^{2} + 28206 T^{4} - 136 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 592 T^{2} + 144510 T^{4} - 592 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 112 T^{2} + 118878 T^{4} - 112 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 436 T^{2} + 275334 T^{4} - 436 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 + 76 p T^{2} + 1290726 T^{4} + 76 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 46 T + 1698 T^{2} + 46 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 1540 T^{2} + 1506054 T^{4} - 1540 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4720 T^{2} + 9299454 T^{4} - 4720 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 64 T + 2334 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 + 4868 T^{2} + 12528486 T^{4} + 4868 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 5540 T^{2} + 15331014 T^{4} + 5540 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 10480 T^{2} + 43220094 T^{4} - 10480 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 2744 T^{2} + 25695534 T^{4} + 2744 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 38 T + p^{2} T^{2} )^{8} \) |
| 67 | \( ( 1 + 8564 T^{2} + 44158854 T^{4} + 8564 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 3028 T^{2} - 19410330 T^{4} - 3028 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 17140 T^{2} + 129420582 T^{4} - 17140 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 22660 T^{2} + 205335174 T^{4} - 22660 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 23492 T^{2} + 230783910 T^{4} + 23492 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 140 T + 19830 T^{2} - 140 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 + 8252 T^{2} + 163205766 T^{4} + 8252 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.85515723896523060687612931322, −6.83431869619299086821250699683, −6.76390497164608425220034557678, −6.72641944700508647733971289322, −6.16088186568913454431046374600, −6.06231484544028540828553644927, −5.90506249402014908128425721758, −5.56387263243683371047532774365, −5.29072090516213096373086466641, −5.09574977081002269468318713733, −5.06245046059594834524209294761, −4.96697171481084920974518127229, −4.94337561389568052467302108314, −4.35863472850823854302830513567, −4.05306091365102585056042690753, −3.91882163760871580695074459189, −3.72819777884412453162754378647, −3.37123330768847173465987819710, −3.36052077497821543269114841007, −3.16758847432878325446883541284, −2.24036874985020074972194857390, −1.86973123275540396521949781418, −1.78676721456718306322720195618, −1.77107311513187937728737987597, −0.39000399673529603641987553530,
0.39000399673529603641987553530, 1.77107311513187937728737987597, 1.78676721456718306322720195618, 1.86973123275540396521949781418, 2.24036874985020074972194857390, 3.16758847432878325446883541284, 3.36052077497821543269114841007, 3.37123330768847173465987819710, 3.72819777884412453162754378647, 3.91882163760871580695074459189, 4.05306091365102585056042690753, 4.35863472850823854302830513567, 4.94337561389568052467302108314, 4.96697171481084920974518127229, 5.06245046059594834524209294761, 5.09574977081002269468318713733, 5.29072090516213096373086466641, 5.56387263243683371047532774365, 5.90506249402014908128425721758, 6.06231484544028540828553644927, 6.16088186568913454431046374600, 6.72641944700508647733971289322, 6.76390497164608425220034557678, 6.83431869619299086821250699683, 6.85515723896523060687612931322
Plot not available for L-functions of degree greater than 10.
