| L(s) = 1 | + 2·2-s − 4-s − 10·5-s − 2·7-s − 20·10-s − 2·11-s − 38·13-s − 4·14-s + 12·16-s − 58·17-s + 10·20-s − 4·22-s + 62·23-s + 25·25-s − 76·26-s + 2·28-s − 48·29-s − 52·31-s + 26·32-s − 116·34-s + 20·35-s − 26·37-s + 52·41-s − 26·43-s + 2·44-s + 124·46-s + 44·47-s + ⋯ |
| L(s) = 1 | + 2-s − 1/4·4-s − 2·5-s − 2/7·7-s − 2·10-s − 0.181·11-s − 2.92·13-s − 2/7·14-s + 3/4·16-s − 3.41·17-s + 1/2·20-s − 0.181·22-s + 2.69·23-s + 25-s − 2.92·26-s + 1/14·28-s − 1.65·29-s − 1.67·31-s + 0.812·32-s − 3.41·34-s + 4/7·35-s − 0.702·37-s + 1.26·41-s − 0.604·43-s + 1/22·44-s + 2.69·46-s + 0.936·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.09523519506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09523519506\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + 5 T^{2} - 3 p^{2} T^{3} + 17 T^{4} - 3 p^{4} T^{5} + 5 p^{4} T^{6} - p^{7} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 2 T + 122 T^{2} + 408 T^{3} + 7487 T^{4} + 408 p^{2} T^{5} + 122 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - 212 T^{2} - 52 T^{3} + 31531 T^{4} - 52 p^{2} T^{5} - 212 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 38 T + 761 T^{2} + 10458 T^{3} + 134144 T^{4} + 10458 p^{2} T^{5} + 761 p^{4} T^{6} + 38 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 58 T + 1682 T^{2} + 34104 T^{3} + 602087 T^{4} + 34104 p^{2} T^{5} + 1682 p^{4} T^{6} + 58 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 508 T^{2} + 106458 T^{4} - 508 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 62 T + 1250 T^{2} + 5088 T^{3} - 606433 T^{4} + 5088 p^{2} T^{5} + 1250 p^{4} T^{6} - 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 48 T + 2417 T^{2} + 79152 T^{3} + 2657808 T^{4} + 79152 p^{2} T^{5} + 2417 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 52 T + 1078 T^{2} - 15392 T^{3} - 878189 T^{4} - 15392 p^{2} T^{5} + 1078 p^{4} T^{6} + 52 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 26 T + 338 T^{2} + 9360 T^{3} - 758881 T^{4} + 9360 p^{2} T^{5} + 338 p^{4} T^{6} + 26 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 52 T - 22 p T^{2} - 12688 T^{3} + 5632051 T^{4} - 12688 p^{2} T^{5} - 22 p^{5} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 26 T + 2978 T^{2} + 140076 T^{3} + 6075743 T^{4} + 140076 p^{2} T^{5} + 2978 p^{4} T^{6} + 26 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 44 T + 584 T^{2} + 131616 T^{3} - 8494033 T^{4} + 131616 p^{2} T^{5} + 584 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 88 T + 3872 T^{2} + 213576 T^{3} + 11634734 T^{4} + 213576 p^{2} T^{5} + 3872 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 156 T + 17066 T^{2} + 1396824 T^{3} + 96294627 T^{4} + 1396824 p^{2} T^{5} + 17066 p^{4} T^{6} + 156 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 2 T - 6467 T^{2} + 1942 T^{3} + 28013116 T^{4} + 1942 p^{2} T^{5} - 6467 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 70 T + 2066 T^{2} + 456156 T^{3} - 37761985 T^{4} + 456156 p^{2} T^{5} + 2066 p^{4} T^{6} - 70 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 134 T + 13248 T^{2} - 134 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 158 T + 12482 T^{2} + 974544 T^{3} + 75384527 T^{4} + 974544 p^{2} T^{5} + 12482 p^{4} T^{6} + 158 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 138 T + 17168 T^{2} - 1493160 T^{3} + 117740187 T^{4} - 1493160 p^{2} T^{5} + 17168 p^{4} T^{6} - 138 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 16448 T^{2} - 120720 T^{3} + 129941999 T^{4} - 120720 p^{2} T^{5} + 16448 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 1102 T^{2} - 98160717 T^{4} - 1102 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 134 T + 24938 T^{2} + 27900 p T^{3} + 33551 p^{2} T^{4} + 27900 p^{3} T^{5} + 24938 p^{4} T^{6} + 134 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−7.73453786948798309441819333349, −7.72675596258492529563789316288, −7.43678697923827233477749568896, −7.25856867309469429920728193840, −6.84314593269163728525287751064, −6.77676453025344222148498688318, −6.62543609046065506846090599639, −6.21368929658699127408501783975, −5.76623223304469519928320454034, −5.45521118350249632217178480361, −5.06517993872242692280563626328, −4.85841554234899866840802031731, −4.66284070177115489446035806065, −4.65888314410242019860087627878, −4.47427532409369062143686666112, −3.84954784093900852657594931011, −3.66277185782574290073304972367, −3.57743120909753394194440187429, −3.07824934786688140310244373808, −2.59265231510924277495717589200, −2.44223281338416046811072796693, −1.96938839030454432726272047771, −1.54575483569711302883503612692, −0.48247456626948490408548070858, −0.099065668301956841509608932043,
0.099065668301956841509608932043, 0.48247456626948490408548070858, 1.54575483569711302883503612692, 1.96938839030454432726272047771, 2.44223281338416046811072796693, 2.59265231510924277495717589200, 3.07824934786688140310244373808, 3.57743120909753394194440187429, 3.66277185782574290073304972367, 3.84954784093900852657594931011, 4.47427532409369062143686666112, 4.65888314410242019860087627878, 4.66284070177115489446035806065, 4.85841554234899866840802031731, 5.06517993872242692280563626328, 5.45521118350249632217178480361, 5.76623223304469519928320454034, 6.21368929658699127408501783975, 6.62543609046065506846090599639, 6.77676453025344222148498688318, 6.84314593269163728525287751064, 7.25856867309469429920728193840, 7.43678697923827233477749568896, 7.72675596258492529563789316288, 7.73453786948798309441819333349
