| L(s) = 1 | − 3-s + 2·4-s + 5·5-s + 6·7-s + 11-s − 2·12-s + 13-s − 5·15-s + 16·17-s + 7·19-s + 10·20-s − 6·21-s − 23-s + 10·25-s + 12·28-s + 16·29-s + 2·31-s − 33-s + 30·35-s − 32·37-s − 39-s − 9·41-s + 16·43-s + 2·44-s − 5·47-s + 27·49-s − 16·51-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 4-s + 2.23·5-s + 2.26·7-s + 0.301·11-s − 0.577·12-s + 0.277·13-s − 1.29·15-s + 3.88·17-s + 1.60·19-s + 2.23·20-s − 1.30·21-s − 0.208·23-s + 2·25-s + 2.26·28-s + 2.97·29-s + 0.359·31-s − 0.174·33-s + 5.07·35-s − 5.26·37-s − 0.160·39-s − 1.40·41-s + 2.43·43-s + 0.301·44-s − 0.729·47-s + 27/7·49-s − 2.24·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.57931161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.57931161\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 - 6 T + 9 T^{2} + 38 T^{3} - 201 T^{4} + 38 p T^{5} + 9 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - T + 3 T^{2} + 25 T^{3} + 56 T^{4} + 25 p T^{5} + 3 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 8 T + 45 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 7 T + 50 T^{2} - 257 T^{3} + 1489 T^{4} - 257 p T^{5} + 50 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + T - 7 T^{2} - 95 T^{3} + 236 T^{4} - 95 p T^{5} - 7 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 16 T + 107 T^{2} - 558 T^{3} + 3125 T^{4} - 558 p T^{5} + 107 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - 2 T - 27 T^{2} + 116 T^{3} + 605 T^{4} + 116 p T^{5} - 27 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 16 T + 133 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 9 T - 10 T^{2} - 339 T^{3} - 1601 T^{4} - 339 p T^{5} - 10 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 5 T - 37 T^{2} - 5 p T^{3} + 824 T^{4} - 5 p^{2} T^{5} - 37 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 59 | $D_{4}$ | \( ( 1 + 20 T + 213 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 25 T + 204 T^{2} + 215 T^{3} - 5149 T^{4} + 215 p T^{5} + 204 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_4\times C_2$ | \( 1 + 5 T - 52 T^{2} + 25 T^{3} + 4849 T^{4} + 25 p T^{5} - 52 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 3 T + 38 T^{2} + 501 T^{3} + 6805 T^{4} + 501 p T^{5} + 38 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 16 T + 63 T^{2} - 470 T^{3} + 8141 T^{4} - 470 p T^{5} + 63 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 13 T + 169 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 13 T + 177 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2:C_4$ | \( 1 + 21 T + 82 T^{2} - 987 T^{3} - 13625 T^{4} - 987 p T^{5} + 82 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 11 T + 223 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.30715369646643356032789915563, −7.20063646940200341787915966382, −6.98384717976928165369185897973, −6.51303545684797073458793771004, −6.41553762690720333253450638443, −5.98560738096690531346684329708, −5.92420016616336094397533923318, −5.78927319689414320375667086601, −5.29561492991666005217397824211, −5.29148242770070408001745545153, −5.22915021965242356227065277091, −5.00790806883718892467541774961, −4.75345540913791061985839800891, −4.08396791058112937070642651717, −4.06347714690742462519845512961, −3.70996254469413544827851899597, −3.14485114188570152656481869364, −2.87786821272035556738974369001, −2.87308016406743266356679183633, −2.55559208582337973731620566705, −1.77593525111813581967611952764, −1.68406895566555451440730513698, −1.44194495152659055117292577154, −1.34968471524381952200865201550, −0.891531798662345719147331095131,
0.891531798662345719147331095131, 1.34968471524381952200865201550, 1.44194495152659055117292577154, 1.68406895566555451440730513698, 1.77593525111813581967611952764, 2.55559208582337973731620566705, 2.87308016406743266356679183633, 2.87786821272035556738974369001, 3.14485114188570152656481869364, 3.70996254469413544827851899597, 4.06347714690742462519845512961, 4.08396791058112937070642651717, 4.75345540913791061985839800891, 5.00790806883718892467541774961, 5.22915021965242356227065277091, 5.29148242770070408001745545153, 5.29561492991666005217397824211, 5.78927319689414320375667086601, 5.92420016616336094397533923318, 5.98560738096690531346684329708, 6.41553762690720333253450638443, 6.51303545684797073458793771004, 6.98384717976928165369185897973, 7.20063646940200341787915966382, 7.30715369646643356032789915563
