| L(s) = 1 | + 4-s − 10·7-s − 3·9-s + 14·13-s + 4·25-s − 10·28-s − 3·36-s − 8·37-s − 2·43-s + 61·49-s + 14·52-s + 42·61-s + 30·63-s − 64-s + 14·67-s − 40·79-s − 140·91-s − 60·97-s + 4·100-s + 64·109-s − 42·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.77·7-s − 9-s + 3.88·13-s + 4/5·25-s − 1.88·28-s − 1/2·36-s − 1.31·37-s − 0.304·43-s + 61/7·49-s + 1.94·52-s + 5.37·61-s + 3.77·63-s − 1/8·64-s + 1.71·67-s − 4.50·79-s − 14.6·91-s − 6.09·97-s + 2/5·100-s + 6.13·109-s − 3.88·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.733661796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.733661796\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 31 T^{2} + 672 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 22 T^{2} - 357 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - 34 T^{2} - 525 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 115 T^{2} + 9744 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 21 T + 208 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - 83 T^{2} + 1848 T^{4} - 83 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 65 T^{2} - 3696 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 30 T + 397 T^{2} + 30 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.970850716534487869591728533369, −7.21759981637035214836016169394, −7.16005700038761164970458985962, −6.97072219524592442835947530235, −6.76509423165465274267035370772, −6.54773845643728969585196784119, −6.40423691233855354695161131347, −6.27467289039538312441668662926, −5.72672409780619261455166829805, −5.63605488360140158124573566538, −5.57538352715981009243147289681, −5.51717491006573136966926618960, −4.86041018093877842648602834444, −4.15245959246954650254325836108, −3.95462694815001455169349720256, −3.86774433149422269300402792838, −3.75974620136415396465409957550, −3.25755924531291235987304875989, −3.01588620857011542329596182404, −2.80728756873877499829968158059, −2.78053215209773970123267891374, −1.96716567404590414797394638490, −1.59236153932022501320565971414, −0.818037648525389861386538676562, −0.56425753180822568154598412060,
0.56425753180822568154598412060, 0.818037648525389861386538676562, 1.59236153932022501320565971414, 1.96716567404590414797394638490, 2.78053215209773970123267891374, 2.80728756873877499829968158059, 3.01588620857011542329596182404, 3.25755924531291235987304875989, 3.75974620136415396465409957550, 3.86774433149422269300402792838, 3.95462694815001455169349720256, 4.15245959246954650254325836108, 4.86041018093877842648602834444, 5.51717491006573136966926618960, 5.57538352715981009243147289681, 5.63605488360140158124573566538, 5.72672409780619261455166829805, 6.27467289039538312441668662926, 6.40423691233855354695161131347, 6.54773845643728969585196784119, 6.76509423165465274267035370772, 6.97072219524592442835947530235, 7.16005700038761164970458985962, 7.21759981637035214836016169394, 7.970850716534487869591728533369
