| L(s) = 1 | + 254·9-s + 444·11-s − 3.40e3·23-s − 298·25-s + 2.07e4·29-s + 1.37e4·37-s + 2.89e4·43-s + 528·53-s − 1.95e5·67-s − 2.16e4·71-s + 1.17e5·79-s − 1.20e4·81-s + 1.12e5·99-s − 2.69e5·107-s + 2.89e5·109-s − 8.64e4·113-s − 4.05e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7.40e5·169-s + ⋯ |
| L(s) = 1 | + 1.04·9-s + 1.10·11-s − 1.34·23-s − 0.0953·25-s + 4.57·29-s + 1.64·37-s + 2.39·43-s + 0.0258·53-s − 5.31·67-s − 0.510·71-s + 2.11·79-s − 0.203·81-s + 1.15·99-s − 2.27·107-s + 2.33·109-s − 0.636·113-s − 2.51·121-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 − 1.99·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.934244969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.934244969\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 254 T^{2} + 8506 p^{2} T^{4} - 254 p^{10} T^{6} + p^{20} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 298 T^{2} + 6577026 T^{4} + 298 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 222 T + 276750 T^{2} - 222 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 740098 T^{2} + 285471304674 T^{4} + 740098 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2066816 T^{2} + 3492088932162 T^{4} - 2066816 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 7905234 T^{2} + 26965772526314 T^{4} + 7905234 p^{10} T^{6} + p^{20} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 1704 T + 9907518 T^{2} + 1704 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 10362 T + 65039082 T^{2} - 10362 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 84821428 T^{2} + 3232208172412950 T^{4} + 84821428 p^{10} T^{6} + p^{20} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 6866 T + 108429786 T^{2} - 6866 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 142394512 T^{2} + 26662994997130146 T^{4} + 142394512 p^{10} T^{6} + p^{20} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 14498 T + 117660750 T^{2} - 14498 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 589410580 T^{2} + 178163045986440726 T^{4} + 589410580 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 264 T + 825104502 T^{2} - 264 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 1953103714 T^{2} + 1882648297019141514 T^{4} + 1953103714 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 58683750 T^{2} + 647470282450909154 T^{4} - 58683750 p^{10} T^{6} + p^{20} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 97692 T + 4619030630 T^{2} + 97692 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 10836 T + 3286930894 T^{2} + 10836 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 1948268208 T^{2} + 9053299624842383906 T^{4} + 1948268208 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 58652 T + 4470747774 T^{2} - 58652 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 9338114914 T^{2} + 42792027293328418122 T^{4} + 9338114914 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 17355789760 T^{2} + \)\(13\!\cdots\!02\)\( T^{4} + 17355789760 p^{10} T^{6} + p^{20} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 33557013216 T^{2} + \)\(42\!\cdots\!10\)\( T^{4} + 33557013216 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−6.52897561037627444468554703337, −6.26584088602572726356797325398, −6.17179344643558329263562972030, −6.01694760039519927622429990959, −5.97406284070736939913274617091, −5.44216357872796234196502551338, −5.00206936414679590518825903066, −4.98447086544110312602331464006, −4.59266406479237992641535047284, −4.35440155718264946677635244194, −4.24254585814476366165236946168, −4.13752564794700705023683681250, −3.86843400481386852952851890279, −3.38880418197122650482583575837, −3.07624201199237541957761594002, −2.89291838437584190381658835076, −2.63894914338290846557212132588, −2.24574659604893835092873375820, −2.22326553825315008480747927908, −1.45307887881390729921887484298, −1.38089144553797206850479095209, −1.24042248246654867813933168934, −0.843339614166417512697578810747, −0.65535975778137028839418837245, −0.17143741740942468476368882576,
0.17143741740942468476368882576, 0.65535975778137028839418837245, 0.843339614166417512697578810747, 1.24042248246654867813933168934, 1.38089144553797206850479095209, 1.45307887881390729921887484298, 2.22326553825315008480747927908, 2.24574659604893835092873375820, 2.63894914338290846557212132588, 2.89291838437584190381658835076, 3.07624201199237541957761594002, 3.38880418197122650482583575837, 3.86843400481386852952851890279, 4.13752564794700705023683681250, 4.24254585814476366165236946168, 4.35440155718264946677635244194, 4.59266406479237992641535047284, 4.98447086544110312602331464006, 5.00206936414679590518825903066, 5.44216357872796234196502551338, 5.97406284070736939913274617091, 6.01694760039519927622429990959, 6.17179344643558329263562972030, 6.26584088602572726356797325398, 6.52897561037627444468554703337
