| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 2·5-s + 4·6-s − 2·7-s + 2·8-s + 9-s − 4·10-s + 6·11-s − 2·12-s + 4·14-s − 4·15-s − 4·16-s − 2·18-s + 4·19-s + 2·20-s + 4·21-s − 12·22-s + 2·23-s − 4·24-s + 11·25-s + 2·27-s − 2·28-s − 20·29-s + 8·30-s + 2·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 1.63·6-s − 0.755·7-s + 0.707·8-s + 1/3·9-s − 1.26·10-s + 1.80·11-s − 0.577·12-s + 1.06·14-s − 1.03·15-s − 16-s − 0.471·18-s + 0.917·19-s + 0.447·20-s + 0.872·21-s − 2.55·22-s + 0.417·23-s − 0.816·24-s + 11/5·25-s + 0.384·27-s − 0.377·28-s − 3.71·29-s + 1.46·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8941917605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8941917605\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 32 T^{2} + 735 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T - 24 T^{2} - 8 T^{3} + 935 T^{4} - 8 p T^{5} - 24 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 10 T + 65 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2 T - 27 T^{2} + 2 p T^{3} - 4 p T^{4} + 2 p^{2} T^{5} - 27 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T - 12 T^{2} + 184 T^{3} - 1177 T^{4} + 184 p T^{5} - 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 100 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T + 46 T^{2} - 496 T^{3} - 2061 T^{4} - 496 p T^{5} + 46 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 10 T - 23 T^{2} - 170 T^{3} + 7020 T^{4} - 170 p T^{5} - 23 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10 T - 25 T^{2} - 70 T^{3} + 6244 T^{4} - 70 p T^{5} - 25 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 20 T + 198 T^{2} - 1360 T^{3} + 9515 T^{4} - 1360 p T^{5} + 198 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 60 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 114 T^{2} + 7667 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 2 T - 153 T^{2} - 2 T^{3} + 18092 T^{4} - 2 p T^{5} - 153 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 2 T + 39 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 4 T - 4 T^{2} - 632 T^{3} - 8945 T^{4} - 632 p T^{5} - 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 10 T + 217 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−7.21884720926094081766596283022, −6.99734403473974909215378474809, −6.77807674926413520475378799927, −6.56485399361678175725032078656, −6.32120569169977036863396451410, −6.09097071538205884548705388954, −5.72063167563025534286538157232, −5.67090338692064512379778593471, −5.50581256005384880812755183741, −5.29317422911654618217618339367, −4.87358620062194285953328035100, −4.79306308806139794463628573103, −4.24118741443182359269470920858, −4.09544115576368824465427532432, −3.87961509407812953181201397515, −3.61371532190862804853129675936, −3.42107707822748857855838296284, −2.71901848258866962937637022354, −2.70341631340370001113556135879, −2.29490781854906930480249292512, −1.95764570977693090815243267452, −1.28847866547259278671972320792, −1.26940145688420037913210240131, −0.866809536525105088644741031593, −0.41796732134421822642017607744,
0.41796732134421822642017607744, 0.866809536525105088644741031593, 1.26940145688420037913210240131, 1.28847866547259278671972320792, 1.95764570977693090815243267452, 2.29490781854906930480249292512, 2.70341631340370001113556135879, 2.71901848258866962937637022354, 3.42107707822748857855838296284, 3.61371532190862804853129675936, 3.87961509407812953181201397515, 4.09544115576368824465427532432, 4.24118741443182359269470920858, 4.79306308806139794463628573103, 4.87358620062194285953328035100, 5.29317422911654618217618339367, 5.50581256005384880812755183741, 5.67090338692064512379778593471, 5.72063167563025534286538157232, 6.09097071538205884548705388954, 6.32120569169977036863396451410, 6.56485399361678175725032078656, 6.77807674926413520475378799927, 6.99734403473974909215378474809, 7.21884720926094081766596283022
