| L(s) = 1 | + 2-s + 4·3-s − 3·5-s + 4·6-s + 4·7-s + 10·9-s − 3·10-s − 2·11-s + 4·13-s + 4·14-s − 12·15-s − 16-s − 2·17-s + 10·18-s + 7·19-s + 16·21-s − 2·22-s + 3·23-s − 25-s + 4·26-s + 20·27-s + 29-s − 12·30-s + 3·31-s − 32-s − 8·33-s − 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.30·3-s − 1.34·5-s + 1.63·6-s + 1.51·7-s + 10/3·9-s − 0.948·10-s − 0.603·11-s + 1.10·13-s + 1.06·14-s − 3.09·15-s − 1/4·16-s − 0.485·17-s + 2.35·18-s + 1.60·19-s + 3.49·21-s − 0.426·22-s + 0.625·23-s − 1/5·25-s + 0.784·26-s + 3.84·27-s + 0.185·29-s − 2.19·30-s + 0.538·31-s − 0.176·32-s − 1.39·33-s − 0.342·34-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
| \(L(1)\) |
\(\approx\) |
\(6.336376542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.336376542\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - T + T^{2} - T^{3} + p T^{4} - p T^{5} + p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 3 T + 2 p T^{2} + p^{2} T^{3} + 74 T^{4} + p^{3} T^{5} + 2 p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2 T + 20 T^{2} + 34 T^{3} + 294 T^{4} + 34 p T^{5} + 20 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 2 T + 40 T^{2} + 62 T^{3} + 878 T^{4} + 62 p T^{5} + 40 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 7 T + 64 T^{2} - 351 T^{3} + 1774 T^{4} - 351 p T^{5} + 64 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 3 T + 40 T^{2} + 49 T^{3} + 494 T^{4} + 49 p T^{5} + 40 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - T + 86 T^{2} - 35 T^{3} + 3378 T^{4} - 35 p T^{5} + 86 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 3 T - 4 T^{2} - 119 T^{3} + 58 p T^{4} - 119 p T^{5} - 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 10 T + 64 T^{2} - 270 T^{3} + 1870 T^{4} - 270 p T^{5} + 64 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 16 T + 4 p T^{2} + 1280 T^{3} + 8694 T^{4} + 1280 p T^{5} + 4 p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 3 T + 128 T^{2} - 275 T^{3} + 7246 T^{4} - 275 p T^{5} + 128 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 5 T + 148 T^{2} - 689 T^{3} + 9638 T^{4} - 689 p T^{5} + 148 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 5 T + 174 T^{2} - 727 T^{3} + 12802 T^{4} - 727 p T^{5} + 174 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 20 T + 316 T^{2} + 3236 T^{3} + 28790 T^{4} + 3236 p T^{5} + 316 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 12 T + 180 T^{2} - 1508 T^{3} + 15014 T^{4} - 1508 p T^{5} + 180 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 22 T + 228 T^{2} + 1254 T^{3} + 6086 T^{4} + 1254 p T^{5} + 228 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 52 T^{2} + 304 T^{3} + 7478 T^{4} + 304 p T^{5} + 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 13 T + 126 T^{2} - 261 T^{3} - 3934 T^{4} - 261 p T^{5} + 126 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 11 T + 196 T^{2} - 1167 T^{3} + 15030 T^{4} - 1167 p T^{5} + 196 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - T + 296 T^{2} - 329 T^{3} + 35310 T^{4} - 329 p T^{5} + 296 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 5 T + 194 T^{2} + 139 T^{3} + 16986 T^{4} + 139 p T^{5} + 194 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 17 T + 374 T^{2} + 4127 T^{3} + 52210 T^{4} + 4127 p T^{5} + 374 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.665553729300073867917416106964, −8.207223953076670859631588495581, −8.140897107322258190302295179480, −8.090083866786603322353590624015, −7.59812598639803922425882860881, −7.59725820643528041546968534191, −7.31828630396094412973955876395, −7.07590021453430569261566645838, −6.64240652262746016190324874885, −6.48750008931052689296052616965, −5.79326950310041801455857676961, −5.66972635446158703455868932222, −5.41369456039447429979351854216, −4.83973347330053675956896965071, −4.47083612524782670720743209297, −4.43190800443986335732644538989, −4.42163015107767800794202903827, −3.68341177093957050134082259414, −3.62387459756726331630318828744, −3.17005981994733725362425823749, −2.96959373929293272112849619855, −2.55677068828348872473276058447, −2.06303734463239796355681587821, −1.53473120245872328500297646797, −1.13463035130736646229008723762,
1.13463035130736646229008723762, 1.53473120245872328500297646797, 2.06303734463239796355681587821, 2.55677068828348872473276058447, 2.96959373929293272112849619855, 3.17005981994733725362425823749, 3.62387459756726331630318828744, 3.68341177093957050134082259414, 4.42163015107767800794202903827, 4.43190800443986335732644538989, 4.47083612524782670720743209297, 4.83973347330053675956896965071, 5.41369456039447429979351854216, 5.66972635446158703455868932222, 5.79326950310041801455857676961, 6.48750008931052689296052616965, 6.64240652262746016190324874885, 7.07590021453430569261566645838, 7.31828630396094412973955876395, 7.59725820643528041546968534191, 7.59812598639803922425882860881, 8.090083866786603322353590624015, 8.140897107322258190302295179480, 8.207223953076670859631588495581, 8.665553729300073867917416106964
