| L(s) = 1 | − 8·11-s − 20·29-s + 12·31-s + 28·41-s − 2·49-s + 20·59-s − 12·61-s − 8·71-s + 2·81-s − 60·89-s + 28·101-s + 20·109-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2.41·11-s − 3.71·29-s + 2.15·31-s + 4.37·41-s − 2/7·49-s + 2.60·59-s − 1.53·61-s − 0.949·71-s + 2/9·81-s − 6.35·89-s + 2.78·101-s + 1.91·109-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05906018712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05906018712\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 1363 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 130 T^{2} + 7603 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 250 T^{2} + 24523 T^{4} - 250 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 101 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 8638 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 240 T^{2} + 27998 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 30 T + 398 T^{2} + 30 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 360 T^{2} + 51038 T^{4} - 360 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.05616830021155433746266937816, −5.98479716112337461782433213735, −5.70883862085279828867330562576, −5.66521753198278851820034462698, −5.47942157351040370949303088165, −5.45920589556857131676084899165, −4.89403423048427344367072439043, −4.87552626524980226773160730441, −4.52485772829227384204003152822, −4.38524926996349585037815827913, −4.13144706755480360444829486602, −4.06221795813139764516577667612, −3.81232072359085965644846244769, −3.30912315600369660256099493713, −3.11771625041123512885857908301, −3.04953882838238640811421960804, −2.77160492027785213635822520288, −2.46606666472666573872426815570, −2.24949752254809086456957177626, −2.00832736100600395018865565229, −1.92631480164631027206484601485, −1.27178433283752652936136872820, −1.00440390636227903252348483389, −0.68986670435369074269108093697, −0.04413071265307839223630961030,
0.04413071265307839223630961030, 0.68986670435369074269108093697, 1.00440390636227903252348483389, 1.27178433283752652936136872820, 1.92631480164631027206484601485, 2.00832736100600395018865565229, 2.24949752254809086456957177626, 2.46606666472666573872426815570, 2.77160492027785213635822520288, 3.04953882838238640811421960804, 3.11771625041123512885857908301, 3.30912315600369660256099493713, 3.81232072359085965644846244769, 4.06221795813139764516577667612, 4.13144706755480360444829486602, 4.38524926996349585037815827913, 4.52485772829227384204003152822, 4.87552626524980226773160730441, 4.89403423048427344367072439043, 5.45920589556857131676084899165, 5.47942157351040370949303088165, 5.66521753198278851820034462698, 5.70883862085279828867330562576, 5.98479716112337461782433213735, 6.05616830021155433746266937816
