| L(s) = 1 | − 17·4-s − 18·9-s − 44·11-s + 113·16-s − 100·19-s + 396·29-s + 720·31-s + 306·36-s − 1.56e3·41-s + 748·44-s + 308·49-s + 344·59-s − 1.55e3·61-s − 17·64-s + 1.26e3·71-s + 1.70e3·76-s − 1.30e3·79-s + 243·81-s + 1.51e3·89-s + 792·99-s + 1.30e3·101-s − 684·109-s − 6.73e3·116-s + 1.21e3·121-s − 1.22e4·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2.12·4-s − 2/3·9-s − 1.20·11-s + 1.76·16-s − 1.20·19-s + 2.53·29-s + 4.17·31-s + 1.41·36-s − 5.95·41-s + 2.56·44-s + 0.897·49-s + 0.759·59-s − 3.26·61-s − 0.0332·64-s + 2.10·71-s + 2.56·76-s − 1.85·79-s + 1/3·81-s + 1.80·89-s + 0.804·99-s + 1.28·101-s − 0.601·109-s − 5.38·116-s + 0.909·121-s − 8.86·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1142235561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1142235561\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 17 T^{2} + 11 p^{4} T^{4} + 17 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 44 p T^{2} + 35526 T^{4} - 44 p^{7} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8144 T^{2} + 26147502 T^{4} - 8144 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4528 T^{2} - 3874 T^{4} - 4528 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 50 T + 14246 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 34840 T^{2} + 555984878 T^{4} - 34840 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 198 T + 57706 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 360 T + 90430 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 17676 T^{2} - 1844934986 T^{4} - 17676 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 782 T + 285970 T^{2} + 782 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 210744 T^{2} + 21303493054 T^{4} - 210744 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 114328 T^{2} + 15430252046 T^{4} - 114328 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 317840 T^{2} + 50316127566 T^{4} - 317840 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 172 T + 175654 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 778 T + 577250 T^{2} + 778 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 454988 T^{2} + 98091174486 T^{4} - 454988 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 630 T + 744334 T^{2} - 630 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 678236 T^{2} + 385736611110 T^{4} - 678236 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 652 T + 589506 T^{2} + 652 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1054364 T^{2} + 869850231414 T^{4} - 1054364 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 756 T + 1427110 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1760636 T^{2} + 2258265821574 T^{4} - 1760636 p^{6} T^{6} + p^{12} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.87090082648338106374639347948, −6.69026628224795576758126242600, −6.44196288977741245650399220509, −6.40352955534580610646980517930, −5.94489986311442455734048861629, −5.61211940066337639298633050077, −5.55016951689399035500583943240, −5.09460488016520987726379059902, −4.96587645635245544888778507983, −4.68908931992931874560240375029, −4.54822864837259315271261709495, −4.46336118178523952277968438202, −4.38865190526260566825593154124, −3.62446093411829798716442687730, −3.56347057291526263799893827187, −3.19057215278539379595302908791, −3.03332406419574166703074924222, −2.59600989687639132322466372467, −2.56682912881532821197533110756, −1.85713428132662860627649155592, −1.81393487503975013913538260158, −1.15557914797694368133464218840, −0.65509718880976434817584810235, −0.63986573779988583038275854889, −0.07088090112064358365883675183,
0.07088090112064358365883675183, 0.63986573779988583038275854889, 0.65509718880976434817584810235, 1.15557914797694368133464218840, 1.81393487503975013913538260158, 1.85713428132662860627649155592, 2.56682912881532821197533110756, 2.59600989687639132322466372467, 3.03332406419574166703074924222, 3.19057215278539379595302908791, 3.56347057291526263799893827187, 3.62446093411829798716442687730, 4.38865190526260566825593154124, 4.46336118178523952277968438202, 4.54822864837259315271261709495, 4.68908931992931874560240375029, 4.96587645635245544888778507983, 5.09460488016520987726379059902, 5.55016951689399035500583943240, 5.61211940066337639298633050077, 5.94489986311442455734048861629, 6.40352955534580610646980517930, 6.44196288977741245650399220509, 6.69026628224795576758126242600, 6.87090082648338106374639347948
