L(s) = 1 | + 2·2-s − 2·3-s + 4-s + 5-s − 4·6-s + 2·7-s − 2·8-s + 9-s + 2·10-s − 6·11-s − 2·12-s + 4·13-s + 4·14-s − 2·15-s − 4·16-s − 3·17-s + 2·18-s − 2·19-s + 20-s − 4·21-s − 12·22-s + 2·23-s + 4·24-s + 6·25-s + 8·26-s + 2·27-s + 2·28-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s − 1.63·6-s + 0.755·7-s − 0.707·8-s + 1/3·9-s + 0.632·10-s − 1.80·11-s − 0.577·12-s + 1.10·13-s + 1.06·14-s − 0.516·15-s − 16-s − 0.727·17-s + 0.471·18-s − 0.458·19-s + 0.223·20-s − 0.872·21-s − 2.55·22-s + 0.417·23-s + 0.816·24-s + 6/5·25-s + 1.56·26-s + 0.384·27-s + 0.377·28-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
\(L(1)\) |
\(\approx\) |
\(2.253959330\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.253959330\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
good | 5 | $D_4\times C_2$ | \( 1 - T - p T^{2} + 4 T^{3} + 6 T^{4} + 4 p T^{5} - p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_4$ | \( ( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 3 T + 11 T^{2} - 108 T^{3} - 438 T^{4} - 108 p T^{5} + 11 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 2 T - 18 T^{2} - 32 T^{3} + 47 T^{4} - 32 p T^{5} - 18 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 9 T + 37 T^{2} - 162 T^{3} - 1536 T^{4} - 162 p T^{5} + 37 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 10 T + 18 T^{2} + 80 T^{3} + 1655 T^{4} + 80 p T^{5} + 18 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 5 T - 71 T^{2} - 10 T^{3} + 5832 T^{4} - 10 p T^{5} - 71 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 3 T - 95 T^{2} - 6 T^{3} + 7530 T^{4} - 6 p T^{5} - 95 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 7 T - 77 T^{2} + 56 T^{3} + 9504 T^{4} + 56 p T^{5} - 77 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 5 T + 144 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 5 T + 81 T^{2} - 1010 T^{3} - 4018 T^{4} - 1010 p T^{5} + 81 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 5 T^{2} - 6216 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 9 T + 182 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 110 T^{2} + 4179 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 11 T + 186 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.01251738225130809958189297114, −6.98337200124481083298114474751, −6.90047060177718909233198804692, −6.15710070658001845151138145240, −6.02265462178953227187472094271, −5.98802587236872535905884804597, −5.86766842697337339331618211020, −5.37951881501475106518104942936, −5.25265086422370415470923845168, −5.17343921830887270756629001507, −5.17176358291732217979618909008, −4.56344153807107556674003014013, −4.42757315066369881782894070906, −4.33658161393679135672321912256, −3.95168450934028951524638683919, −3.57725880496013843522034315143, −3.26008737015181042572538406773, −3.12168029932276558979264617427, −3.04631096637876966198096573958, −2.32157030391958868786276447015, −1.96099803363809023124931215359, −1.85384854996235994534110290183, −1.67817850869430104443043320724, −0.66463016408914392829976673251, −0.40519644280904451749285082641,
0.40519644280904451749285082641, 0.66463016408914392829976673251, 1.67817850869430104443043320724, 1.85384854996235994534110290183, 1.96099803363809023124931215359, 2.32157030391958868786276447015, 3.04631096637876966198096573958, 3.12168029932276558979264617427, 3.26008737015181042572538406773, 3.57725880496013843522034315143, 3.95168450934028951524638683919, 4.33658161393679135672321912256, 4.42757315066369881782894070906, 4.56344153807107556674003014013, 5.17176358291732217979618909008, 5.17343921830887270756629001507, 5.25265086422370415470923845168, 5.37951881501475106518104942936, 5.86766842697337339331618211020, 5.98802587236872535905884804597, 6.02265462178953227187472094271, 6.15710070658001845151138145240, 6.90047060177718909233198804692, 6.98337200124481083298114474751, 7.01251738225130809958189297114