| L(s) = 1 | − 8·5-s + 12·9-s + 24·13-s − 8·17-s + 4·25-s − 136·29-s − 40·37-s + 8·41-s − 96·45-s + 100·49-s + 88·53-s − 296·61-s − 192·65-s + 88·73-s + 74·81-s + 64·85-s + 216·89-s − 328·97-s + 24·101-s + 440·109-s − 312·113-s + 288·117-s + 140·121-s + 72·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 8/5·5-s + 4/3·9-s + 1.84·13-s − 0.470·17-s + 4/25·25-s − 4.68·29-s − 1.08·37-s + 8/41·41-s − 2.13·45-s + 2.04·49-s + 1.66·53-s − 4.85·61-s − 2.95·65-s + 1.20·73-s + 0.913·81-s + 0.752·85-s + 2.42·89-s − 3.38·97-s + 0.237·101-s + 4.03·109-s − 2.76·113-s + 2.46·117-s + 1.15·121-s + 0.575·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.071190043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.071190043\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 p T^{2} + 70 T^{4} - 4 p^{5} T^{6} + p^{8} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 4 T + 22 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 5254 T^{4} - 100 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 5382 T^{4} - 140 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 12 T + 342 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 454 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 780 T^{2} + 402374 T^{4} - 780 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 484 T^{2} + 517894 T^{4} - 484 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 68 T + 2806 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 20 T + 1270 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 2854 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2764 T^{2} + 8710534 T^{4} - 2764 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 7428 T^{2} + 23258246 T^{4} - 7428 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 44 T + 6070 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 11020 T^{2} + 52720774 T^{4} - 11020 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 148 T + 11350 T^{2} + 148 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 11980 T^{2} + 75342534 T^{4} - 11980 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18276 T^{2} + 133865606 T^{4} - 18276 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 44 T + 9990 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 1028 T^{2} + 28324230 T^{4} - 1028 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 5452 T^{2} + 97966918 T^{4} - 5452 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 108 T + 15558 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 164 T + 22342 T^{2} + 164 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 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
−9.482415242292395898999986522455, −9.134491802445198085943160626560, −9.008809517576601987174247640977, −8.904409043717117786516135233944, −8.465614376984604626953553328724, −7.968288357792941920740897658606, −7.85589822902003476388861024287, −7.44080832153534550443785301186, −7.33208117860013835690844846885, −7.29518255921657874472388963989, −6.72476594789313867635975437160, −6.25578985257200962034051496173, −6.09285948987201867634232653981, −5.64095651069235511338251980481, −5.42796951014303270055757795129, −4.94912889505918606651902993562, −4.36091838568098770719405034547, −4.13114686178600122343947459149, −3.87530572858742689251081683986, −3.53371627007778845925567966444, −3.47819599012845139278189130025, −2.52485474928293717583848819731, −1.71513495582123759050072940933, −1.60492484920060555566738344747, −0.42420625848744924549397251279,
0.42420625848744924549397251279, 1.60492484920060555566738344747, 1.71513495582123759050072940933, 2.52485474928293717583848819731, 3.47819599012845139278189130025, 3.53371627007778845925567966444, 3.87530572858742689251081683986, 4.13114686178600122343947459149, 4.36091838568098770719405034547, 4.94912889505918606651902993562, 5.42796951014303270055757795129, 5.64095651069235511338251980481, 6.09285948987201867634232653981, 6.25578985257200962034051496173, 6.72476594789313867635975437160, 7.29518255921657874472388963989, 7.33208117860013835690844846885, 7.44080832153534550443785301186, 7.85589822902003476388861024287, 7.968288357792941920740897658606, 8.465614376984604626953553328724, 8.904409043717117786516135233944, 9.008809517576601987174247640977, 9.134491802445198085943160626560, 9.482415242292395898999986522455
