| L(s) = 1 | + 8·9-s − 32·25-s − 16·37-s + 12·49-s + 30·81-s + 80·109-s + 88·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 256·225-s + ⋯ |
| L(s) = 1 | + 8/3·9-s − 6.39·25-s − 2.63·37-s + 12/7·49-s + 10/3·81-s + 7.66·109-s + 8·121-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 + 4.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s − 17.0·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.172505069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.172505069\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - p T^{2} )^{8} \) |
| 13 | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 108 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 112 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + p T^{2} )^{8} \) |
| 71 | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 156 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 166 T^{2} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.55888734645072786957060115004, −4.42345455345556853429540150794, −4.31475816764735676458320920034, −4.08803300965341202633061791277, −4.04049920909941846092705967937, −3.88718694007484185309563535294, −3.82362419969975428280808564303, −3.48549609534085610928150575054, −3.48504064832037719158625230986, −3.36511544133394812280678750625, −3.33818609167608772701135247177, −3.28046025678454824584829070319, −2.71653512446063766474166715921, −2.62638965509510593224722374341, −2.39382602341030359544141286169, −2.16933294829108335598226306306, −1.99254757192471304373775414012, −1.95134804043058088338021627838, −1.81349429963779302689789288564, −1.80306740227037151909350174751, −1.45977132553862462958433519583, −1.34033437428094932240817782453, −0.72076197732311380538598137953, −0.65437565394123676918291992286, −0.33789247658355296638885217871,
0.33789247658355296638885217871, 0.65437565394123676918291992286, 0.72076197732311380538598137953, 1.34033437428094932240817782453, 1.45977132553862462958433519583, 1.80306740227037151909350174751, 1.81349429963779302689789288564, 1.95134804043058088338021627838, 1.99254757192471304373775414012, 2.16933294829108335598226306306, 2.39382602341030359544141286169, 2.62638965509510593224722374341, 2.71653512446063766474166715921, 3.28046025678454824584829070319, 3.33818609167608772701135247177, 3.36511544133394812280678750625, 3.48504064832037719158625230986, 3.48549609534085610928150575054, 3.82362419969975428280808564303, 3.88718694007484185309563535294, 4.04049920909941846092705967937, 4.08803300965341202633061791277, 4.31475816764735676458320920034, 4.42345455345556853429540150794, 4.55888734645072786957060115004
Plot not available for L-functions of degree greater than 10.
