| L(s) = 1 | + 3·3-s − 6·5-s + 8·7-s + 2·9-s + 6·11-s − 6·13-s − 18·15-s − 9·17-s + 24·21-s + 17·25-s − 3·27-s + 12·29-s − 3·31-s + 18·33-s − 48·35-s − 30·37-s − 18·39-s − 16·43-s − 12·45-s + 18·47-s + 34·49-s − 27·51-s − 12·53-s − 36·55-s − 15·59-s − 21·61-s + 16·63-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 2.68·5-s + 3.02·7-s + 2/3·9-s + 1.80·11-s − 1.66·13-s − 4.64·15-s − 2.18·17-s + 5.23·21-s + 17/5·25-s − 0.577·27-s + 2.22·29-s − 0.538·31-s + 3.13·33-s − 8.11·35-s − 4.93·37-s − 2.88·39-s − 2.43·43-s − 1.78·45-s + 2.62·47-s + 34/7·49-s − 3.78·51-s − 1.64·53-s − 4.85·55-s − 1.95·59-s − 2.68·61-s + 2.01·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.534942640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.534942640\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + p T^{2} )^{2}( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 9 T + 35 T^{2} + 108 T^{3} + 450 T^{4} + 108 p T^{5} + 35 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 5 T^{2} - 336 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 13 T^{2} - 360 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 3 T + 19 T^{2} - 216 T^{3} - 1140 T^{4} - 216 p T^{5} + 19 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 15 T + 112 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 145 T^{2} + 8544 T^{4} - 145 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 - 18 T + 185 T^{2} - 1386 T^{3} + 8796 T^{4} - 1386 p T^{5} + 185 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 12 T + 67 T^{2} + 228 T^{3} + 96 T^{4} + 228 p T^{5} + 67 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + 15 T + p T^{2} )^{2}( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $D_4\times C_2$ | \( 1 + 21 T + 281 T^{2} + 2814 T^{3} + 23202 T^{4} + 2814 p T^{5} + 281 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - T - 59 T^{2} + 74 T^{3} - 956 T^{4} + 74 p T^{5} - 59 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 15 T + 31 T^{2} - 720 T^{3} + 15882 T^{4} - 720 p T^{5} + 31 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 15 T + 227 T^{2} - 2280 T^{3} + 22788 T^{4} - 2280 p T^{5} + 227 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 256 T^{2} + 28974 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 9 T + 209 T^{2} + 1638 T^{3} + 27606 T^{4} + 1638 p T^{5} + 209 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
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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
−7.87992139976569971252983677683, −7.75049716394205296545017759202, −7.26348537544585052039243760772, −7.19569835422291053836148471317, −7.03358834232389918778517675241, −6.75442253110355250185376604552, −6.61396137118206583193000964219, −6.24931676259091064277203067345, −5.60926204163872716876869807868, −5.50123657131675100320349625054, −4.86890309093460796137302937704, −4.82833738354876528807852985373, −4.75134454382759077070910168191, −4.53567288710972915180151236986, −4.18697626311757296010366888002, −4.02456592811304267507678057125, −3.61578201001503181297810900749, −3.32055851164108050144722219273, −3.16409121746690453604652509910, −2.82114814482273217016360053502, −2.21841482439727375486448472061, −1.96082522732491280287430236980, −1.73229560501111931033302257452, −1.35498314607043636232034577428, −0.32826091338237338288021241635,
0.32826091338237338288021241635, 1.35498314607043636232034577428, 1.73229560501111931033302257452, 1.96082522732491280287430236980, 2.21841482439727375486448472061, 2.82114814482273217016360053502, 3.16409121746690453604652509910, 3.32055851164108050144722219273, 3.61578201001503181297810900749, 4.02456592811304267507678057125, 4.18697626311757296010366888002, 4.53567288710972915180151236986, 4.75134454382759077070910168191, 4.82833738354876528807852985373, 4.86890309093460796137302937704, 5.50123657131675100320349625054, 5.60926204163872716876869807868, 6.24931676259091064277203067345, 6.61396137118206583193000964219, 6.75442253110355250185376604552, 7.03358834232389918778517675241, 7.19569835422291053836148471317, 7.26348537544585052039243760772, 7.75049716394205296545017759202, 7.87992139976569971252983677683
