| L(s) = 1 | + 2·2-s + 2·3-s + 4-s + 4·6-s + 2·7-s − 2·8-s + 9-s + 2·11-s + 2·12-s + 6·13-s + 4·14-s − 4·16-s + 2·17-s + 2·18-s − 6·19-s + 4·21-s + 4·22-s + 2·23-s − 4·24-s + 12·26-s − 2·27-s + 2·28-s + 6·29-s + 16·31-s − 2·32-s + 4·33-s + 4·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.63·6-s + 0.755·7-s − 0.707·8-s + 1/3·9-s + 0.603·11-s + 0.577·12-s + 1.66·13-s + 1.06·14-s − 16-s + 0.485·17-s + 0.471·18-s − 1.37·19-s + 0.872·21-s + 0.852·22-s + 0.417·23-s − 0.816·24-s + 2.35·26-s − 0.384·27-s + 0.377·28-s + 1.11·29-s + 2.87·31-s − 0.353·32-s + 0.696·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(19.20648865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.20648865\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 2 T - 8 T^{2} + 4 T^{3} + 67 T^{4} + 4 p T^{5} - 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 4 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T - 19 T^{2} + 22 T^{3} + 172 T^{4} + 22 p T^{5} - 19 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T - 8 T^{2} + 36 T^{3} + 891 T^{4} + 36 p T^{5} - 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 52 T^{3} - 221 T^{4} + 52 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T - 19 T^{2} + 18 T^{3} + 1140 T^{4} + 18 p T^{5} - 19 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2 T - 23 T^{2} + 94 T^{3} - 788 T^{4} + 94 p T^{5} - 23 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 8 T - 7 T^{2} + 88 T^{3} + 736 T^{4} + 88 p T^{5} - 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T - 8 T^{2} + 220 T^{3} + 5515 T^{4} + 220 p T^{5} - 8 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 139 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4 T + 2 T^{2} + 416 T^{3} - 4229 T^{4} + 416 p T^{5} + 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T + 37 T^{2} - 572 T^{3} - 4256 T^{4} - 572 p T^{5} + 37 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 2 T - 56 T^{2} - 148 T^{3} - 1157 T^{4} - 148 p T^{5} - 56 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T - 112 T^{2} + 36 T^{3} + 14307 T^{4} + 36 p T^{5} - 112 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 8 T + 87 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 + 14 T + 212 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 4 T - 118 T^{2} - 176 T^{3} + 8611 T^{4} - 176 p T^{5} - 118 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.40021508065832168982600818412, −6.36203185160931943859186275328, −6.22979378631966223500489896304, −5.75335477443050846171578513844, −5.70499125722044162871030313679, −5.64475712212537717246223483151, −5.06470395111084144661073976653, −4.90769610475150645729936164571, −4.72592463605759589454873799262, −4.58128951033937909682282506261, −4.32625133698592226968269237333, −4.20153470649733930268900162412, −3.87976489547270539604290728719, −3.72471667165049509052633006044, −3.59286117092053039049998725854, −2.96821224236244426968215427557, −2.91846206950680049868949572002, −2.89446939770363416851386006523, −2.74422042506076475094974567877, −2.16650057307795582869863523196, −1.74336562978274807672317519686, −1.72032277675907784967917181176, −1.28911704080273454261365840138, −0.820771662027204888197696014781, −0.57354921231064176360029293585,
0.57354921231064176360029293585, 0.820771662027204888197696014781, 1.28911704080273454261365840138, 1.72032277675907784967917181176, 1.74336562978274807672317519686, 2.16650057307795582869863523196, 2.74422042506076475094974567877, 2.89446939770363416851386006523, 2.91846206950680049868949572002, 2.96821224236244426968215427557, 3.59286117092053039049998725854, 3.72471667165049509052633006044, 3.87976489547270539604290728719, 4.20153470649733930268900162412, 4.32625133698592226968269237333, 4.58128951033937909682282506261, 4.72592463605759589454873799262, 4.90769610475150645729936164571, 5.06470395111084144661073976653, 5.64475712212537717246223483151, 5.70499125722044162871030313679, 5.75335477443050846171578513844, 6.22979378631966223500489896304, 6.36203185160931943859186275328, 6.40021508065832168982600818412
