| L(s) = 1 | − 8·5-s + 4·7-s − 2·9-s + 8·11-s − 8·13-s + 7·16-s + 8·17-s + 12·19-s + 32·25-s − 8·29-s + 12·31-s − 32·35-s + 12·37-s + 16·41-s + 16·45-s − 16·47-s − 2·49-s − 8·53-s − 64·55-s + 8·59-s − 8·63-s + 64·65-s + 4·67-s − 12·73-s + 32·77-s + 8·79-s − 56·80-s + ⋯ |
| L(s) = 1 | − 3.57·5-s + 1.51·7-s − 2/3·9-s + 2.41·11-s − 2.21·13-s + 7/4·16-s + 1.94·17-s + 2.75·19-s + 32/5·25-s − 1.48·29-s + 2.15·31-s − 5.40·35-s + 1.97·37-s + 2.49·41-s + 2.38·45-s − 2.33·47-s − 2/7·49-s − 1.09·53-s − 8.62·55-s + 1.04·59-s − 1.00·63-s + 7.93·65-s + 0.488·67-s − 1.40·73-s + 3.64·77-s + 0.900·79-s − 6.26·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.352686977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.352686977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - 7 T^{4} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 56 T^{3} + 82 T^{4} - 56 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 19 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 300 T^{3} + 1214 T^{4} - 300 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 442 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 444 T^{3} + 2702 T^{4} - 444 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 84 T^{3} - 802 T^{4} - 84 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6678 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1072 T^{3} + 8578 T^{4} + 1072 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 440 T^{3} + 6034 T^{4} - 440 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 156 T^{3} - 8194 T^{4} + 156 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 + 6818 T^{4} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 516 T^{3} + 2798 T^{4} + 516 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 632 T^{3} + 12466 T^{4} - 632 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 804 T^{3} + 8078 T^{4} + 804 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.329737683479690016816681381942, −8.066805716109816110699993791443, −7.979219896661870137615300253636, −7.893673005435995003922091007169, −7.87963834668404501310736882369, −7.55869524182978500225695167800, −7.40554456387400213538618322281, −7.01867158655110493116961113647, −6.57197364372148838592487478230, −6.55282958961159371465570313225, −5.98241657034860454174584029530, −5.46801939783081782476974494643, −5.26529520665939108739305043002, −5.25441587395853182134093092199, −4.72416777117522853757758542864, −4.41707444395358256015728101616, −4.07671566487956019322798460834, −3.87328209562343459947022186901, −3.78216503385497611169377832204, −3.05217263645611403260673527893, −2.93464523081225141131066306959, −2.91523800453843093926981502307, −1.55380507360430923360558612234, −1.18640937313623032494966389139, −0.75275537138415115216343060825,
0.75275537138415115216343060825, 1.18640937313623032494966389139, 1.55380507360430923360558612234, 2.91523800453843093926981502307, 2.93464523081225141131066306959, 3.05217263645611403260673527893, 3.78216503385497611169377832204, 3.87328209562343459947022186901, 4.07671566487956019322798460834, 4.41707444395358256015728101616, 4.72416777117522853757758542864, 5.25441587395853182134093092199, 5.26529520665939108739305043002, 5.46801939783081782476974494643, 5.98241657034860454174584029530, 6.55282958961159371465570313225, 6.57197364372148838592487478230, 7.01867158655110493116961113647, 7.40554456387400213538618322281, 7.55869524182978500225695167800, 7.87963834668404501310736882369, 7.893673005435995003922091007169, 7.979219896661870137615300253636, 8.066805716109816110699993791443, 8.329737683479690016816681381942
