| L(s) = 1 | + 8·2-s + 18·3-s + 16·4-s + 108·5-s + 144·6-s − 128·8-s + 81·9-s + 864·10-s − 124·11-s + 288·12-s − 1.44e3·13-s + 1.94e3·15-s − 1.02e3·16-s − 1.26e3·17-s + 648·18-s + 360·19-s + 1.72e3·20-s − 992·22-s − 6.52e3·23-s − 2.30e3·24-s + 6.71e3·25-s − 1.15e4·26-s − 1.45e3·27-s + 1.41e4·29-s + 1.55e4·30-s + 5.90e3·31-s − 2.04e3·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.93·5-s + 1.63·6-s − 0.707·8-s + 1/3·9-s + 2.73·10-s − 0.308·11-s + 0.577·12-s − 2.36·13-s + 2.23·15-s − 16-s − 1.05·17-s + 0.471·18-s + 0.228·19-s + 0.965·20-s − 0.436·22-s − 2.57·23-s − 0.816·24-s + 2.14·25-s − 3.34·26-s − 0.384·27-s + 3.13·29-s + 3.15·30-s + 1.10·31-s − 0.353·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(12.87804478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.87804478\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 108 T + 4948 T^{2} - 50328 T^{3} - 1110969 T^{4} - 50328 p^{5} T^{5} + 4948 p^{10} T^{6} - 108 p^{15} T^{7} + p^{20} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 124 T - 278818 T^{2} - 3460592 T^{3} + 58136364859 T^{4} - 3460592 p^{5} T^{5} - 278818 p^{10} T^{6} + 124 p^{15} T^{7} + p^{20} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 720 T + 673736 T^{2} + 720 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 1260 T - 1209092 T^{2} - 54207720 T^{3} + 3551327269215 T^{4} - 54207720 p^{5} T^{5} - 1209092 p^{10} T^{6} + 1260 p^{15} T^{7} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 360 T - 4647630 T^{2} + 62988480 T^{3} + 16369957784699 T^{4} + 62988480 p^{5} T^{5} - 4647630 p^{10} T^{6} - 360 p^{15} T^{7} + p^{20} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6524 T + 19335014 T^{2} + 2937183088 p T^{3} + 423716043763 p^{2} T^{4} + 2937183088 p^{6} T^{5} + 19335014 p^{10} T^{6} + 6524 p^{15} T^{7} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 7088 T + 48216146 T^{2} - 7088 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 5904 T - 23971190 T^{2} - 9269894016 T^{3} + 1643220565216419 T^{4} - 9269894016 p^{5} T^{5} - 23971190 p^{10} T^{6} - 5904 p^{15} T^{7} + p^{20} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 6040 T - 74621402 T^{2} + 166612868480 T^{3} + 5005514179302955 T^{4} + 166612868480 p^{5} T^{5} - 74621402 p^{10} T^{6} - 6040 p^{15} T^{7} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 17388 T + 216792980 T^{2} - 17388 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 608 T + 164053110 T^{2} + 608 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 648 p T + 298147738 T^{2} + 110633159232 p T^{3} + 130837578639633603 T^{4} + 110633159232 p^{6} T^{5} + 298147738 p^{10} T^{6} + 648 p^{16} T^{7} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 3964 T - 578718526 T^{2} - 959126126096 T^{3} + 171890491073351707 T^{4} - 959126126096 p^{5} T^{5} - 578718526 p^{10} T^{6} + 3964 p^{15} T^{7} + p^{20} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 40752 T - 180305678 T^{2} - 16756512663168 T^{3} + 1690986043017054027 T^{4} - 16756512663168 p^{5} T^{5} - 180305678 p^{10} T^{6} - 40752 p^{15} T^{7} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 1368 T - 1543594872 T^{2} - 196617586608 T^{3} + 1673542562621074391 T^{4} - 196617586608 p^{5} T^{5} - 1543594872 p^{10} T^{6} + 1368 p^{15} T^{7} + p^{20} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16224 T - 549961574 T^{2} + 30615831207936 T^{3} - 1516953966778043781 T^{4} + 30615831207936 p^{5} T^{5} - 549961574 p^{10} T^{6} - 16224 p^{15} T^{7} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 3204 T - 212201462 T^{2} + 3204 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 23976 T + 506144208 T^{2} + 97760673100368 T^{3} - 5484607792703379793 T^{4} + 97760673100368 p^{5} T^{5} + 506144208 p^{10} T^{6} - 23976 p^{15} T^{7} + p^{20} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 1040 p T - 1035403070 T^{2} - 1696818106880 p T^{3} + 30377412596963147299 T^{4} - 1696818106880 p^{6} T^{5} - 1035403070 p^{10} T^{6} - 1040 p^{16} T^{7} + p^{20} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 173736 T + 14732805782 T^{2} + 173736 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 200556 T + 19064326876 T^{2} - 2003607258829272 T^{3} + \)\(19\!\cdots\!35\)\( T^{4} - 2003607258829272 p^{5} T^{5} + 19064326876 p^{10} T^{6} - 200556 p^{15} T^{7} + p^{20} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 251928 T + 32348722272 T^{2} + 251928 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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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
−7.76098142906066108877135266947, −7.29433711899035757315862484277, −7.08979245573586115360182728680, −6.80950400419099369812627654849, −6.61545259779112006506340454457, −6.13166889015983905626727900124, −5.94286353236680425702889475061, −5.86227944470570397242242344409, −5.74366767397527999003523320786, −5.06355014619340073476629186913, −4.84891800199306761638731825166, −4.72409667515195608625694652029, −4.42972253595802301193796804279, −4.38179243061646611440402596918, −3.68956975617950421620194180398, −3.61278462043272959497986302497, −2.81591379628559282954867351180, −2.79476834306928668934256448948, −2.58413574235771668159232023638, −2.34006760858001028025383182342, −2.15926633740515517148271558999, −1.67471991784309015324619986436, −1.16551238210768898715563918903, −0.64467589223355718018740164248, −0.29408986605787766245766384009,
0.29408986605787766245766384009, 0.64467589223355718018740164248, 1.16551238210768898715563918903, 1.67471991784309015324619986436, 2.15926633740515517148271558999, 2.34006760858001028025383182342, 2.58413574235771668159232023638, 2.79476834306928668934256448948, 2.81591379628559282954867351180, 3.61278462043272959497986302497, 3.68956975617950421620194180398, 4.38179243061646611440402596918, 4.42972253595802301193796804279, 4.72409667515195608625694652029, 4.84891800199306761638731825166, 5.06355014619340073476629186913, 5.74366767397527999003523320786, 5.86227944470570397242242344409, 5.94286353236680425702889475061, 6.13166889015983905626727900124, 6.61545259779112006506340454457, 6.80950400419099369812627654849, 7.08979245573586115360182728680, 7.29433711899035757315862484277, 7.76098142906066108877135266947
