| L(s) = 1 | − 162·9-s + 128·11-s − 5.28e3·19-s − 3.88e3·29-s + 1.26e4·31-s + 4.80e3·41-s + 1.94e4·49-s − 1.03e4·59-s − 1.07e5·61-s − 1.43e5·71-s − 5.85e4·79-s + 1.96e4·81-s + 2.79e5·89-s − 2.07e4·99-s − 1.00e5·101-s − 2.48e4·109-s + 1.28e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.78e5·169-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 0.318·11-s − 3.35·19-s − 0.856·29-s + 2.36·31-s + 0.446·41-s + 1.15·49-s − 0.387·59-s − 3.71·61-s − 3.36·71-s − 1.05·79-s + 1/3·81-s + 3.73·89-s − 0.212·99-s − 0.975·101-s − 0.200·109-s + 0.797·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.02·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.990735659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.990735659\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 19452 T^{2} + 653446630 T^{4} - 19452 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 64 T - 5278 p T^{2} - 64 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 378892 T^{2} + 21095389878 T^{4} - 378892 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2146780 T^{2} + 2395021035398 T^{4} - 2146780 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 2640 T + 5527222 T^{2} + 2640 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 20045156 T^{2} + 183268460410982 T^{4} + 20045156 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 1940 T + 17567422 T^{2} + 1940 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 6328 T + 63242942 T^{2} - 6328 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 212769516 T^{2} + 20570809389128662 T^{4} - 212769516 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2404 T - 34529258 T^{2} - 2404 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2098380 T^{2} + 4366275807638998 T^{4} - 2098380 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 161331324 T^{2} + 93283739073012038 T^{4} - 161331324 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1302086668 T^{2} + 752861101325672150 T^{4} - 1302086668 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 5184 T + 1405691158 T^{2} + 5184 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 53956 T + 1627858910 T^{2} + 53956 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 503801516 T^{2} + 2038466458659863862 T^{4} - 503801516 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 71552 T + 3857659342 T^{2} + 71552 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 3629392476 T^{2} + 11739219546601563238 T^{4} - 3629392476 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 29288 T + 5671921950 T^{2} + 29288 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 3772878572 T^{2} + 16627424423525289078 T^{4} - 3772878572 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 139724 T + 16006792406 T^{2} - 139724 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 28290762620 T^{2} + \)\(34\!\cdots\!98\)\( T^{4} - 28290762620 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−6.76210856812733744793469482779, −6.56809886319925250501344901929, −6.23560757775943356062830494647, −6.21814440460662437111486082010, −6.07222842357958605247297602895, −5.71876858860385753073960349767, −5.63544298962072722362313748365, −4.97261436810029709046749478066, −4.88710479832377492750257334827, −4.70007637495362086866289280637, −4.21078956707216910378183774548, −4.19668915613754132020833983454, −4.11014340641706708122049132462, −3.67362157822989896089421806025, −3.13417418336559793844734134532, −2.96299604677496093885664414843, −2.78189694337142155378575523067, −2.57562050260539505563010473204, −1.92026224120640020717054094414, −1.85710028849166342307203775310, −1.76581930506660553804171429128, −1.12230313700986887762332659604, −0.850263902464793556735348616544, −0.29669844635832454034213775346, −0.28980174167978512633002515247,
0.28980174167978512633002515247, 0.29669844635832454034213775346, 0.850263902464793556735348616544, 1.12230313700986887762332659604, 1.76581930506660553804171429128, 1.85710028849166342307203775310, 1.92026224120640020717054094414, 2.57562050260539505563010473204, 2.78189694337142155378575523067, 2.96299604677496093885664414843, 3.13417418336559793844734134532, 3.67362157822989896089421806025, 4.11014340641706708122049132462, 4.19668915613754132020833983454, 4.21078956707216910378183774548, 4.70007637495362086866289280637, 4.88710479832377492750257334827, 4.97261436810029709046749478066, 5.63544298962072722362313748365, 5.71876858860385753073960349767, 6.07222842357958605247297602895, 6.21814440460662437111486082010, 6.23560757775943356062830494647, 6.56809886319925250501344901929, 6.76210856812733744793469482779
