| L(s) = 1 | − 8·9-s + 72·11-s + 236·19-s + 112·29-s + 40·31-s − 1.08e3·41-s − 98·49-s + 396·59-s − 692·61-s − 2.44e3·71-s − 1.16e3·79-s − 1.18e3·81-s − 1.19e3·89-s − 576·99-s + 7.02e3·101-s − 1.40e3·109-s + 2.02e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.05e3·169-s + ⋯ |
| L(s) = 1 | − 0.296·9-s + 1.97·11-s + 2.84·19-s + 0.717·29-s + 0.231·31-s − 4.14·41-s − 2/7·49-s + 0.873·59-s − 1.45·61-s − 4.09·71-s − 1.66·79-s − 1.62·81-s − 1.41·89-s − 0.584·99-s + 6.91·101-s − 1.23·109-s + 1.51·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.30·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.241067523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.241067523\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 1246 T^{4} + 8 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 36 T + 934 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5056 T^{2} + 13530702 T^{4} - 5056 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 15516 T^{2} + 108330374 T^{4} - 15516 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 118 T + 7566 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 41436 T^{2} + 715443110 T^{4} - 41436 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 56 T + 21974 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 20 T + 48510 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 64204 T^{2} + 4714100022 T^{4} - 64204 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 544 T + 193358 T^{2} + 544 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 210812 T^{2} + 21856770966 T^{4} - 210812 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 267948 T^{2} + 36714463334 T^{4} - 267948 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 31380 T^{2} + 23351768246 T^{4} + 31380 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 198 T + 410926 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 346 T + 429114 T^{2} + 346 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 167884 T^{2} - 79838932650 T^{4} - 167884 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 1224 T + 980014 T^{2} + 1224 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1151996 T^{2} + 596571224550 T^{4} - 1151996 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 584 T + 997470 T^{2} + 584 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1278872 T^{2} + 872265367134 T^{4} - 1278872 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 596 T + 39542 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2725724 T^{2} + 3318683676102 T^{4} - 2725724 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−6.40810194327137737173521499744, −6.30571430592949375214254241118, −6.04953619161145152978535813373, −5.66840752250464213729684667385, −5.63074609161738212300114384082, −5.43710907095316811477599594864, −5.05070536428194863446954068992, −4.89462493296201939464933350389, −4.58263851654938269761484760863, −4.53330816317961141692224388624, −4.24842444026676486656470374308, −3.89125587282130670110780743346, −3.49772882688069006845024295697, −3.41915376843964545069504544980, −3.29525788692032145681823756828, −3.07833857521894440977342454612, −2.63299870780505017505380886694, −2.60435721671910260537692342609, −1.91895429541183252204352800688, −1.62837326130889607811116394652, −1.47433320564953219016087964996, −1.31798557972557275880440614611, −0.998597255824915936325407135391, −0.55305775357214385828268586162, −0.16492319062770170421936353349,
0.16492319062770170421936353349, 0.55305775357214385828268586162, 0.998597255824915936325407135391, 1.31798557972557275880440614611, 1.47433320564953219016087964996, 1.62837326130889607811116394652, 1.91895429541183252204352800688, 2.60435721671910260537692342609, 2.63299870780505017505380886694, 3.07833857521894440977342454612, 3.29525788692032145681823756828, 3.41915376843964545069504544980, 3.49772882688069006845024295697, 3.89125587282130670110780743346, 4.24842444026676486656470374308, 4.53330816317961141692224388624, 4.58263851654938269761484760863, 4.89462493296201939464933350389, 5.05070536428194863446954068992, 5.43710907095316811477599594864, 5.63074609161738212300114384082, 5.66840752250464213729684667385, 6.04953619161145152978535813373, 6.30571430592949375214254241118, 6.40810194327137737173521499744
