| L(s) = 1 | − 2·2-s + 4·3-s + 4-s − 8·6-s + 2·8-s + 11·9-s + 4·12-s + 2·13-s − 4·16-s − 22·18-s − 2·19-s + 2·23-s + 8·24-s + 25-s − 4·26-s + 24·27-s + 2·32-s + 11·36-s + 4·38-s + 8·39-s − 4·46-s − 16·48-s + 49-s − 2·50-s + 2·52-s − 48·54-s − 8·57-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4·3-s + 4-s − 8·6-s + 2·8-s + 11·9-s + 4·12-s + 2·13-s − 4·16-s − 22·18-s − 2·19-s + 2·23-s + 8·24-s + 25-s − 4·26-s + 24·27-s + 2·32-s + 11·36-s + 4·38-s + 8·39-s − 4·46-s − 16·48-s + 49-s − 2·50-s + 2·52-s − 48·54-s − 8·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.668149327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.668149327\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 3 | $C_1$$\times$$C_2^2$ | \( ( 1 - T )^{4}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 43 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_2^2$ | \( ( 1 + T )^{4}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_2^2$ | \( ( 1 - T )^{4}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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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
−6.47289244742668784895682168104, −6.27844209041723390085835423740, −6.06424074198102654227604949706, −5.74062107753476290098831993642, −5.14355284223899592945952557514, −5.09122903636230900385853673630, −4.73953138531396002368187577376, −4.61799984097759015021380845569, −4.58021026108894911067600820193, −4.33765515982474652542828318622, −4.09698832231528588949574882102, −3.96797122715333334081241053111, −3.52572359031910856935543398102, −3.50217306691681596862853724067, −3.41940773689770390549422800124, −3.00763446681404356506278082068, −2.95474803807198056688122884722, −2.35930788952853218228438763049, −2.24357303334499414675160963586, −2.12583250879005686762817406575, −1.93976320731464094318938099064, −1.33164799524263214984465501393, −1.31654905967129161154352117563, −1.11325645100132809737785826950, −1.04954425928130118980189683439,
1.04954425928130118980189683439, 1.11325645100132809737785826950, 1.31654905967129161154352117563, 1.33164799524263214984465501393, 1.93976320731464094318938099064, 2.12583250879005686762817406575, 2.24357303334499414675160963586, 2.35930788952853218228438763049, 2.95474803807198056688122884722, 3.00763446681404356506278082068, 3.41940773689770390549422800124, 3.50217306691681596862853724067, 3.52572359031910856935543398102, 3.96797122715333334081241053111, 4.09698832231528588949574882102, 4.33765515982474652542828318622, 4.58021026108894911067600820193, 4.61799984097759015021380845569, 4.73953138531396002368187577376, 5.09122903636230900385853673630, 5.14355284223899592945952557514, 5.74062107753476290098831993642, 6.06424074198102654227604949706, 6.27844209041723390085835423740, 6.47289244742668784895682168104
