| L(s) = 1 | − 3·2-s − 7·4-s + 15·5-s + 35·8-s − 45·10-s + 11-s + 79·13-s − 19·16-s − 72·17-s + 29·19-s − 105·20-s − 3·22-s + 63·23-s + 150·25-s − 237·26-s − 220·29-s − 136·31-s − 73·32-s + 216·34-s + 43·37-s − 87·38-s + 525·40-s − 599·41-s + 170·43-s − 7·44-s − 189·46-s − 3·47-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 7/8·4-s + 1.34·5-s + 1.54·8-s − 1.42·10-s + 0.0274·11-s + 1.68·13-s − 0.296·16-s − 1.02·17-s + 0.350·19-s − 1.17·20-s − 0.0290·22-s + 0.571·23-s + 6/5·25-s − 1.78·26-s − 1.40·29-s − 0.787·31-s − 0.403·32-s + 1.08·34-s + 0.191·37-s − 0.371·38-s + 2.07·40-s − 2.28·41-s + 0.602·43-s − 0.0239·44-s − 0.605·46-s − 0.00931·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 7 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + 3 T + p^{4} T^{2} + 17 p T^{3} + p^{7} T^{4} + 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - T + 3485 T^{2} - 6442 T^{3} + 3485 p^{3} T^{4} - p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 79 T + 6542 T^{2} - 346439 T^{3} + 6542 p^{3} T^{4} - 79 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 72 T + 14635 T^{2} + 688048 T^{3} + 14635 p^{3} T^{4} + 72 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 29 T + 18484 T^{2} - 332301 T^{3} + 18484 p^{3} T^{4} - 29 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 63 T + 32377 T^{2} - 1479946 T^{3} + 32377 p^{3} T^{4} - 63 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 220 T + 74711 T^{2} + 10703064 T^{3} + 74711 p^{3} T^{4} + 220 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 136 T + 27042 T^{2} + 3338174 T^{3} + 27042 p^{3} T^{4} + 136 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 43 T + 33294 T^{2} - 1889771 T^{3} + 33294 p^{3} T^{4} - 43 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 599 T + 272999 T^{2} + 79436618 T^{3} + 272999 p^{3} T^{4} + 599 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 170 T + 62886 T^{2} - 25979064 T^{3} + 62886 p^{3} T^{4} - 170 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 3 T + 72213 T^{2} + 31379318 T^{3} + 72213 p^{3} T^{4} + 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 331 T + 442671 T^{2} + 98567438 T^{3} + 442671 p^{3} T^{4} + 331 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 1520 T + 1161665 T^{2} + 599760000 T^{3} + 1161665 p^{3} T^{4} + 1520 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 1160 T + 1094643 T^{2} - 571897312 T^{3} + 1094643 p^{3} T^{4} - 1160 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 806 T + 956790 T^{2} - 441427296 T^{3} + 956790 p^{3} T^{4} - 806 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 406 T + 391273 T^{2} - 58294156 T^{3} + 391273 p^{3} T^{4} - 406 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 1192 T + 1524996 T^{2} - 931572678 T^{3} + 1524996 p^{3} T^{4} - 1192 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2590 T + 3310582 T^{2} + 2751421632 T^{3} + 3310582 p^{3} T^{4} + 2590 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 508 T + 1655669 T^{2} + 550491384 T^{3} + 1655669 p^{3} T^{4} + 508 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 42 T + 566539 T^{2} + 649223388 T^{3} + 566539 p^{3} T^{4} - 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 1020 T + 828679 T^{2} + 17707848 T^{3} + 828679 p^{3} T^{4} - 1020 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.344951714692115465036061198064, −7.85585174627434986626154157795, −7.81356264226760211644630947290, −7.29449240384700796080281466753, −6.84947365009698492810905416499, −6.79569451271792971071608831899, −6.64362047069748179244564737741, −6.28719325562379612404547068096, −5.93960916402761307563509344839, −5.61693616682764830050660452102, −5.27737951295699984147133951476, −5.27540294555218759534848978824, −5.06766326493444172788576624384, −4.35558597695686581596506011688, −4.20800304243746528504510158885, −4.17396016224472660594945245478, −3.57112688240831696873163172949, −3.35008191557625810749889035494, −3.04021319832786709087540601030, −2.54737565481397716815772366756, −2.15346551349414905465591295684, −1.90440433540315797185592682805, −1.36472332834112343519432684940, −1.14617529167089068778990737192, −1.09875820679450024195922391510, 0, 0, 0,
1.09875820679450024195922391510, 1.14617529167089068778990737192, 1.36472332834112343519432684940, 1.90440433540315797185592682805, 2.15346551349414905465591295684, 2.54737565481397716815772366756, 3.04021319832786709087540601030, 3.35008191557625810749889035494, 3.57112688240831696873163172949, 4.17396016224472660594945245478, 4.20800304243746528504510158885, 4.35558597695686581596506011688, 5.06766326493444172788576624384, 5.27540294555218759534848978824, 5.27737951295699984147133951476, 5.61693616682764830050660452102, 5.93960916402761307563509344839, 6.28719325562379612404547068096, 6.64362047069748179244564737741, 6.79569451271792971071608831899, 6.84947365009698492810905416499, 7.29449240384700796080281466753, 7.81356264226760211644630947290, 7.85585174627434986626154157795, 8.344951714692115465036061198064
