| L(s) = 1 | + 16·25-s − 16·31-s − 28·49-s + 8·73-s + 48·79-s − 24·97-s + 64·103-s + 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 60·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 16/5·25-s − 2.87·31-s − 4·49-s + 0.936·73-s + 5.40·79-s − 2.43·97-s + 6.30·103-s + 4.36·121-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 + 4.61·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.650779666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.650779666\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 8 T^{2} + 38 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 24 T^{2} + 358 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 30 T^{2} + 451 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 16 T^{2} + 614 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 18 T^{2} + 691 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 16 T^{2} - 250 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 36 T^{2} + 1558 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 37 | \( ( 1 - 102 T^{2} + 4891 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 4 T^{2} + 1574 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 108 T^{2} + 6166 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 80 T^{2} + 3750 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 132 T^{2} + 9526 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 56 T^{2} + 6374 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 46 T^{2} - 1101 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 125 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + p T^{2} )^{8} \) |
| 73 | \( ( 1 - 2 T + 35 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 12 T + 187 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 236 T^{2} + 25910 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 304 T^{2} + 38918 T^{4} + 304 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 6 T + 175 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−4.34368708133810703996311186463, −4.18120994562881468883125064487, −4.16805454259222459642881691746, −3.80120159264971665645909254916, −3.79767811014471942518201921088, −3.77925601199069692367879286941, −3.64704220285438387175083654345, −3.36768107669615223888147751225, −3.33511250783158719637926130006, −3.08380335417315169784650282551, −2.93914083844781911243510315261, −2.87470188038321878037992250783, −2.83501683812561782257844193409, −2.77589944154465405388693319393, −2.21520294651378898806180343797, −2.04073969936277866671011076871, −1.98127223175644880455690087860, −1.89886687837026798273109343862, −1.84776583630653514169988731410, −1.66147211290975114244345675614, −1.23169067231171322230580905508, −0.892304382562949519357214812331, −0.813562671842546427942208295838, −0.75138378105902760509960331117, −0.15972764853230071100038529900,
0.15972764853230071100038529900, 0.75138378105902760509960331117, 0.813562671842546427942208295838, 0.892304382562949519357214812331, 1.23169067231171322230580905508, 1.66147211290975114244345675614, 1.84776583630653514169988731410, 1.89886687837026798273109343862, 1.98127223175644880455690087860, 2.04073969936277866671011076871, 2.21520294651378898806180343797, 2.77589944154465405388693319393, 2.83501683812561782257844193409, 2.87470188038321878037992250783, 2.93914083844781911243510315261, 3.08380335417315169784650282551, 3.33511250783158719637926130006, 3.36768107669615223888147751225, 3.64704220285438387175083654345, 3.77925601199069692367879286941, 3.79767811014471942518201921088, 3.80120159264971665645909254916, 4.16805454259222459642881691746, 4.18120994562881468883125064487, 4.34368708133810703996311186463
Plot not available for L-functions of degree greater than 10.
