| L(s) = 1 | − 2-s + 3-s − 5-s − 6-s − 3·7-s + 3·9-s + 10-s − 11·11-s + 6·13-s + 3·14-s − 15-s + 7·17-s − 3·18-s + 19-s − 3·21-s + 11·22-s − 4·23-s − 6·26-s + 20·29-s + 30-s + 13·31-s + 32-s − 11·33-s − 7·34-s + 3·35-s − 8·37-s − 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.447·5-s − 0.408·6-s − 1.13·7-s + 9-s + 0.316·10-s − 3.31·11-s + 1.66·13-s + 0.801·14-s − 0.258·15-s + 1.69·17-s − 0.707·18-s + 0.229·19-s − 0.654·21-s + 2.34·22-s − 0.834·23-s − 1.17·26-s + 3.71·29-s + 0.182·30-s + 2.33·31-s + 0.176·32-s − 1.91·33-s − 1.20·34-s + 0.507·35-s − 1.31·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.469922489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.469922489\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + p T + 51 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| good | 3 | $C_4\times C_2$ | \( 1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 5 p T^{5} - 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 11 p T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 + 3 T - 3 T^{2} - 5 T^{3} + 36 T^{4} - 5 p T^{5} - 3 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 6 T + 23 T^{2} - 120 T^{3} + 601 T^{4} - 120 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 7 T + 2 T^{2} + 65 T^{3} - 169 T^{4} + 65 p T^{5} + 2 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 - 20 T + 211 T^{2} - 1600 T^{3} + 9561 T^{4} - 1600 p T^{5} + 211 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 13 T + 108 T^{2} - 821 T^{3} + 5525 T^{4} - 821 p T^{5} + 108 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 8 T + 77 T^{2} + 640 T^{3} + 3001 T^{4} + 640 p T^{5} + 77 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 13 T + 53 T^{2} - 331 T^{3} + 3380 T^{4} - 331 p T^{5} + 53 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 17 T + 92 T^{2} - 345 T^{3} + 2621 T^{4} - 345 p T^{5} + 92 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 19 T + 88 T^{2} - 1015 T^{3} - 14829 T^{4} - 1015 p T^{5} + 88 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 10 T - 19 T^{2} + 270 T^{3} + 851 T^{4} + 270 p T^{5} - 19 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 18 T + 263 T^{2} - 2736 T^{3} + 25105 T^{4} - 2736 p T^{5} + 263 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 9 T + 153 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 17 T + 88 T^{2} - 731 T^{3} - 14475 T^{4} - 731 p T^{5} + 88 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - T - 67 T^{2} - 215 T^{3} + 5476 T^{4} - 215 p T^{5} - 67 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 + 15 T + 146 T^{2} + 1005 T^{3} + 3541 T^{4} + 1005 p T^{5} + 146 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 31 T + 383 T^{2} - 2755 T^{3} + 19736 T^{4} - 2755 p T^{5} + 383 p^{2} T^{6} - 31 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 10 T + 23 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 27 T + 282 T^{2} - 2395 T^{3} + 25311 T^{4} - 2395 p T^{5} + 282 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.063519277483862446072754962172, −7.921938183946382418602047827117, −7.69397523502468456027529624980, −7.60654740430458773254282084608, −7.33482016811807422207845765372, −6.91238814191365402185769288964, −6.72808369717639118892167725574, −6.29525711451334411555332160402, −6.10861200329363040806316366733, −5.97240282200856395122011021252, −5.64632014644964590476436907646, −5.37346365615988541958530190702, −4.89672868521433413411625018028, −4.78973988911871631448435624754, −4.44063736686985612816622519737, −4.20492094617288137901934527553, −3.58225279426673764975951211412, −3.55573412870410379632275685279, −3.15460995055558064386582485501, −2.76460849092801612630537335210, −2.54934059175460208798083824939, −2.44866174255394761011154969386, −1.56832581894284171788253727077, −0.855896443166178480029742274647, −0.70317471493933793905103311929,
0.70317471493933793905103311929, 0.855896443166178480029742274647, 1.56832581894284171788253727077, 2.44866174255394761011154969386, 2.54934059175460208798083824939, 2.76460849092801612630537335210, 3.15460995055558064386582485501, 3.55573412870410379632275685279, 3.58225279426673764975951211412, 4.20492094617288137901934527553, 4.44063736686985612816622519737, 4.78973988911871631448435624754, 4.89672868521433413411625018028, 5.37346365615988541958530190702, 5.64632014644964590476436907646, 5.97240282200856395122011021252, 6.10861200329363040806316366733, 6.29525711451334411555332160402, 6.72808369717639118892167725574, 6.91238814191365402185769288964, 7.33482016811807422207845765372, 7.60654740430458773254282084608, 7.69397523502468456027529624980, 7.921938183946382418602047827117, 8.063519277483862446072754962172
