| L(s) = 1 | − 2·2-s + 4-s − 2·5-s − 2·7-s + 2·8-s + 4·10-s − 2·11-s − 16·13-s + 4·14-s − 4·16-s + 10·17-s + 4·19-s − 2·20-s + 4·22-s − 2·23-s + 9·25-s + 32·26-s − 2·28-s − 4·31-s + 2·32-s − 20·34-s + 4·35-s − 8·37-s − 8·38-s − 4·40-s − 4·41-s − 2·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.894·5-s − 0.755·7-s + 0.707·8-s + 1.26·10-s − 0.603·11-s − 4.43·13-s + 1.06·14-s − 16-s + 2.42·17-s + 0.917·19-s − 0.447·20-s + 0.852·22-s − 0.417·23-s + 9/5·25-s + 6.27·26-s − 0.377·28-s − 0.718·31-s + 0.353·32-s − 3.42·34-s + 0.676·35-s − 1.31·37-s − 1.29·38-s − 0.632·40-s − 0.624·41-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2028388238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2028388238\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 2 T - p T^{2} - 2 T^{3} + 36 T^{4} - 2 p T^{5} - p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 10 T + 49 T^{2} - 10 p T^{3} + 36 p T^{4} - 10 p^{2} T^{5} + 49 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T - 18 T^{2} + 16 T^{3} + 491 T^{4} + 16 p T^{5} - 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 2 T - 25 T^{2} - 34 T^{3} + 220 T^{4} - 34 p T^{5} - 25 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_4\times C_2$ | \( 1 + 4 T + 22 T^{2} - 272 T^{3} - 1421 T^{4} - 272 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 6 T^{2} - 128 T^{3} - 373 T^{4} - 128 p T^{5} + 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 10 T - T^{2} - 70 T^{3} + 3292 T^{4} - 70 p T^{5} - p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 94 T^{2} - 896 T^{3} + 9867 T^{4} - 896 p T^{5} + 94 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 4 T - 74 T^{2} - 112 T^{3} + 3675 T^{4} - 112 p T^{5} - 74 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6 T - 93 T^{2} - 42 T^{3} + 10724 T^{4} - 42 p T^{5} - 93 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 10 T + 13 T^{2} + 470 T^{3} - 3620 T^{4} + 470 p T^{5} + 13 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T - 126 T^{2} + 16 T^{3} + 13667 T^{4} + 16 p T^{5} - 126 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 18 T + 103 T^{2} + 1134 T^{3} + 17004 T^{4} + 1134 p T^{5} + 103 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 14 T + 207 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 8 T - 58 T^{2} - 448 T^{3} + 1267 T^{4} - 448 p T^{5} - 58 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 26 T + 355 T^{2} + 26 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.13298533054877379104933178456, −7.02897476909572800431686160951, −6.68078713577740265998364539213, −6.16901182453900406748280458673, −5.86457389759113100252144118018, −5.73394796241594168143685231731, −5.36804199316394466539534649641, −5.21711530540910249436951896769, −5.11056901136977038405331426200, −4.92718593687242269197684456522, −4.87138769082550365315811466773, −4.19206855778397555864052342624, −4.04172649309179217644802879903, −3.90948670734687323650410885090, −3.75773428502498641542591272061, −3.02829206909068152429715690932, −3.02078339964708757512406230601, −2.73141523120162255455525688553, −2.70190915678549223913815716721, −2.24354150718876193339065855400, −1.90973377937653860112327859808, −1.41413047144503022852546804518, −1.04621127453767368454729911395, −0.55407289796744983334991458105, −0.21163091660196047385912949301,
0.21163091660196047385912949301, 0.55407289796744983334991458105, 1.04621127453767368454729911395, 1.41413047144503022852546804518, 1.90973377937653860112327859808, 2.24354150718876193339065855400, 2.70190915678549223913815716721, 2.73141523120162255455525688553, 3.02078339964708757512406230601, 3.02829206909068152429715690932, 3.75773428502498641542591272061, 3.90948670734687323650410885090, 4.04172649309179217644802879903, 4.19206855778397555864052342624, 4.87138769082550365315811466773, 4.92718593687242269197684456522, 5.11056901136977038405331426200, 5.21711530540910249436951896769, 5.36804199316394466539534649641, 5.73394796241594168143685231731, 5.86457389759113100252144118018, 6.16901182453900406748280458673, 6.68078713577740265998364539213, 7.02897476909572800431686160951, 7.13298533054877379104933178456
