| L(s) = 1 | + 2·3-s + 4-s + 9-s − 6·11-s + 2·12-s − 8·17-s + 6·19-s − 2·25-s − 2·27-s + 4·29-s − 12·33-s + 36-s − 24·37-s − 12·43-s − 6·44-s − 5·49-s − 16·51-s − 8·53-s + 12·57-s − 12·59-s + 8·61-s − 64-s + 12·67-s − 8·68-s − 24·71-s − 4·75-s + 6·76-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 1/3·9-s − 1.80·11-s + 0.577·12-s − 1.94·17-s + 1.37·19-s − 2/5·25-s − 0.384·27-s + 0.742·29-s − 2.08·33-s + 1/6·36-s − 3.94·37-s − 1.82·43-s − 0.904·44-s − 5/7·49-s − 2.24·51-s − 1.09·53-s + 1.58·57-s − 1.56·59-s + 1.02·61-s − 1/8·64-s + 1.46·67-s − 0.970·68-s − 2.84·71-s − 0.461·75-s + 0.688·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.517997063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.517997063\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^3$ | \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 3 p T^{2} + 126 T^{3} + 452 T^{4} + 126 p T^{5} + 3 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 37 T^{2} - 150 T^{3} + 492 T^{4} - 150 p T^{5} + 37 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T - 34 T^{2} + 32 T^{3} + 1195 T^{4} + 32 p T^{5} - 34 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 1254 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 24 T + 313 T^{2} + 2904 T^{3} + 20376 T^{4} + 2904 p T^{5} + 313 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 66 T^{2} + 2675 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 146 T^{2} + 9315 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 4 T + 107 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 114 T^{2} + 792 T^{3} + 3707 T^{4} + 792 p T^{5} + 114 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 62 T^{2} - 32 T^{3} + 8251 T^{4} - 32 p T^{5} - 62 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 190 T^{2} - 1704 T^{3} + 18891 T^{4} - 1704 p T^{5} + 190 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 24 T + 346 T^{2} + 3696 T^{3} + 32307 T^{4} + 3696 p T^{5} + 346 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 25974 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 42 T + 909 T^{2} + 13482 T^{3} + 147452 T^{4} + 13482 p T^{5} + 909 p^{2} T^{6} + 42 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 36 T + 730 T^{2} - 10728 T^{3} + 121299 T^{4} - 10728 p T^{5} + 730 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.367222560021834050549049455065, −8.035738755956100543514116898524, −7.84676536867782507686165979142, −7.31326089265622407634235057360, −7.18238653535402705825599364545, −7.05634691366936235753929744769, −7.00738304865843536075069543736, −6.33449532756644432822932937110, −6.24734336895077864804690646787, −6.10766380145351258923259747093, −5.50322319980367032209801563753, −5.25569621345798412674428023677, −5.10929743704409273186439336872, −4.89665574765021678599319375597, −4.46392414259883872720695165795, −4.36347422254600825092272138870, −3.65676584712416392707772064816, −3.27469621266193774831611505546, −3.24575906275120943923344625320, −3.15495164670683074847833400087, −2.42364070266293365675340806923, −2.26903910295893342527651661161, −1.76340583897817762083979213066, −1.70166073836535242231125634679, −0.39663059979433253148458460256,
0.39663059979433253148458460256, 1.70166073836535242231125634679, 1.76340583897817762083979213066, 2.26903910295893342527651661161, 2.42364070266293365675340806923, 3.15495164670683074847833400087, 3.24575906275120943923344625320, 3.27469621266193774831611505546, 3.65676584712416392707772064816, 4.36347422254600825092272138870, 4.46392414259883872720695165795, 4.89665574765021678599319375597, 5.10929743704409273186439336872, 5.25569621345798412674428023677, 5.50322319980367032209801563753, 6.10766380145351258923259747093, 6.24734336895077864804690646787, 6.33449532756644432822932937110, 7.00738304865843536075069543736, 7.05634691366936235753929744769, 7.18238653535402705825599364545, 7.31326089265622407634235057360, 7.84676536867782507686165979142, 8.035738755956100543514116898524, 8.367222560021834050549049455065
