| L(s) = 1 | + 16·4-s + 20·7-s + 408·11-s + 192·16-s + 600·23-s + 2.21e3·25-s + 320·28-s + 1.84e3·29-s + 4.16e3·37-s + 616·43-s + 6.52e3·44-s + 106·49-s − 9.28e3·53-s + 2.04e3·64-s − 1.44e4·67-s + 1.61e4·71-s + 8.16e3·77-s − 1.42e4·79-s + 9.60e3·92-s + 3.53e4·100-s + 3.49e4·107-s − 2.00e4·109-s + 3.84e3·112-s + 3.67e4·113-s + 2.95e4·116-s + 7.37e4·121-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s + 0.408·7-s + 3.37·11-s + 3/4·16-s + 1.13·23-s + 3.53·25-s + 0.408·28-s + 2.19·29-s + 3.04·37-s + 0.333·43-s + 3.37·44-s + 0.0441·49-s − 3.30·53-s + 1/2·64-s − 3.22·67-s + 3.20·71-s + 1.37·77-s − 2.29·79-s + 1.13·92-s + 3.53·100-s + 3.05·107-s − 1.68·109-s + 0.306·112-s + 2.88·113-s + 2.19·116-s + 5.03·121-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(13.53803637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.53803637\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 20 T + 6 p^{2} T^{2} - 20 p^{4} T^{3} + p^{8} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2212 T^{2} + 1986054 T^{4} - 2212 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 204 T + 25574 T^{2} - 204 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 10276 T^{2} - 744216186 T^{4} - 10276 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 290884 T^{2} + 34920570246 T^{4} - 290884 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 278500 T^{2} + 50525353734 T^{4} - 278500 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 300 T + 488870 T^{2} - 300 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 924 T + 1314374 T^{2} - 924 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 178364 T^{2} + 1502662267014 T^{4} + 178364 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 2084 T + 4574886 T^{2} - 2084 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 939076 T^{2} + 14542898957574 T^{4} - 939076 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 308 T + 5284230 T^{2} - 308 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8924932 T^{2} + 64853581075206 T^{4} - 8924932 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 4644 T + 19720838 T^{2} + 4644 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 14882204 T^{2} + 343194815714694 T^{4} + 14882204 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 15427108 T^{2} + 407638379539590 T^{4} - 15427108 p^{8} T^{6} + p^{16} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 7244 T + 42357318 T^{2} + 7244 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 8076 T + 66903014 T^{2} - 8076 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 71407876 T^{2} + 2461031183058054 T^{4} - 71407876 p^{8} T^{6} + p^{16} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 7148 T + 75702246 T^{2} + 7148 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 82408420 T^{2} + 6202361073224454 T^{4} - 82408420 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 109309252 T^{2} + 9513924225311238 T^{4} - 109309252 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 284462596 T^{2} + 35116951743889926 T^{4} - 284462596 p^{8} T^{6} + p^{16} T^{8} \) |
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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
−9.113896200422146128160021461118, −8.825988751491050974114569721443, −8.505598729780622113910222170261, −8.419010380335408761739834965804, −8.019994860609464385361375966325, −7.45231089103884428272674266276, −7.37714071464477132181826053891, −6.81441826532545917155383804790, −6.80793915142786033193364914732, −6.45807017804554660017343140349, −6.36841460932268848217225530261, −5.84172713313933898438109975490, −5.79896961580344752756885010496, −4.75301574472441342839373950025, −4.69695419010637339053263342822, −4.61925345179416995209151632411, −4.23411502796994281525615626910, −3.47648265786769576417383626119, −3.18686256831747594173848213319, −2.98456243357551235329355404798, −2.50136710820609319035970918228, −1.80934706640768310704696609339, −1.24297124472238720027740397665, −0.998501416221369874315244645842, −0.880467529822952917688412952051,
0.880467529822952917688412952051, 0.998501416221369874315244645842, 1.24297124472238720027740397665, 1.80934706640768310704696609339, 2.50136710820609319035970918228, 2.98456243357551235329355404798, 3.18686256831747594173848213319, 3.47648265786769576417383626119, 4.23411502796994281525615626910, 4.61925345179416995209151632411, 4.69695419010637339053263342822, 4.75301574472441342839373950025, 5.79896961580344752756885010496, 5.84172713313933898438109975490, 6.36841460932268848217225530261, 6.45807017804554660017343140349, 6.80793915142786033193364914732, 6.81441826532545917155383804790, 7.37714071464477132181826053891, 7.45231089103884428272674266276, 8.019994860609464385361375966325, 8.419010380335408761739834965804, 8.505598729780622113910222170261, 8.825988751491050974114569721443, 9.113896200422146128160021461118
