| L(s) = 1 | − 16-s + 32·23-s − 48·43-s + 40·53-s + 16·67-s − 80·107-s − 72·109-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 1/4·16-s + 6.67·23-s − 7.31·43-s + 5.49·53-s + 1.95·67-s − 7.73·107-s − 6.89·109-s + 1.09·121-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 + 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 + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.334808016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.334808016\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{4} + p^{4} T^{8} \) |
| 7 | \( ( 1 + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2}( 1 - 206 T^{4} + p^{4} T^{8} ) \) |
| 13 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + p T^{2} )^{8} \) |
| 19 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2}( 1 + 1234 T^{4} + p^{4} T^{8} ) \) |
| 31 | \( ( 1 + p T^{2} )^{8} \) |
| 37 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2}( 1 - 1294 T^{4} + p^{4} T^{8} ) \) |
| 41 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 12 T + p T^{2} )^{4}( 1 - 334 T^{4} + p^{4} T^{8} ) \) |
| 47 | \( ( 1 - p T^{2} )^{8} \) |
| 53 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 - 5582 T^{4} + p^{4} T^{8} ) \) |
| 59 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 4946 T^{4} + p^{4} T^{8} ) \) |
| 71 | \( ( 1 + 2914 T^{4} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 3646 T^{4} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - p T^{2} )^{8} \) |
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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
−5.47625877347543116652107837257, −5.34834782844325160647868507261, −5.33890828626082011841436534888, −5.25308633290917300584645364844, −5.16042019895046236874709084947, −4.75331792737552615199742578861, −4.58447805543994516638845410126, −4.46779481631165699772300407475, −4.36192377352054413270021323128, −4.34575485342817399004896867834, −3.79630310957879825085184628091, −3.74269103107780975951701324299, −3.54325725377358678953532628008, −3.39545982717243189919977493041, −3.14603512836084835542216280942, −3.09001856668160577262137186420, −2.93591725323949068662120431262, −2.52951006492490064765837497611, −2.45827027381107723535448138779, −2.43085086568258066445972264357, −1.81351784080627547355671871761, −1.54202712703220817384561886071, −1.24021335724224941668352710528, −1.20034594838714165189656313292, −0.57107601502338766970294908635,
0.57107601502338766970294908635, 1.20034594838714165189656313292, 1.24021335724224941668352710528, 1.54202712703220817384561886071, 1.81351784080627547355671871761, 2.43085086568258066445972264357, 2.45827027381107723535448138779, 2.52951006492490064765837497611, 2.93591725323949068662120431262, 3.09001856668160577262137186420, 3.14603512836084835542216280942, 3.39545982717243189919977493041, 3.54325725377358678953532628008, 3.74269103107780975951701324299, 3.79630310957879825085184628091, 4.34575485342817399004896867834, 4.36192377352054413270021323128, 4.46779481631165699772300407475, 4.58447805543994516638845410126, 4.75331792737552615199742578861, 5.16042019895046236874709084947, 5.25308633290917300584645364844, 5.33890828626082011841436534888, 5.34834782844325160647868507261, 5.47625877347543116652107837257
Plot not available for L-functions of degree greater than 10.
