| L(s) = 1 | + 4-s − 4·5-s + 9-s + 4·11-s + 10·19-s − 4·20-s + 5·25-s − 40·29-s + 36-s + 4·41-s + 4·44-s − 4·45-s + 2·49-s − 16·55-s + 22·59-s + 24·61-s − 64-s − 12·71-s + 10·76-s − 16·79-s + 10·89-s − 40·95-s + 4·99-s + 5·100-s + 48·101-s + 4·109-s − 40·116-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.78·5-s + 1/3·9-s + 1.20·11-s + 2.29·19-s − 0.894·20-s + 25-s − 7.42·29-s + 1/6·36-s + 0.624·41-s + 0.603·44-s − 0.596·45-s + 2/7·49-s − 2.15·55-s + 2.86·59-s + 3.07·61-s − 1/8·64-s − 1.42·71-s + 1.14·76-s − 1.80·79-s + 1.05·89-s − 4.10·95-s + 0.402·99-s + 1/2·100-s + 4.77·101-s + 0.383·109-s − 3.71·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7624316989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7624316989\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 17 T^{2} + 120 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 102 T^{2} + 7595 T^{4} + 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 34 T^{2} - 3333 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 66 T^{2} - 2533 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 5 T - 64 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
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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
−7.10201365484174287963941947498, −6.94286987516285565539101555076, −6.93382898625983724569592332420, −6.26051792534973769158794296497, −6.16583355125133414259685367787, −5.94752062365395676645696621462, −5.62723625092631929489488712138, −5.40468767264128944277481186789, −5.35002193860007193946744069289, −5.11389822303637143725518061486, −4.78226975012131178775640686162, −4.33702238561373734486307242603, −3.98052925093805615692513134377, −3.88904228834475255816826712674, −3.80971575295479733257891132704, −3.63639334672869551973952218023, −3.30090896050323077553340695764, −3.29496070802326476320960186252, −2.41043263288142054287824514091, −2.40680611933508535784697927084, −2.03914371950139317304233224052, −1.64779610260291178492682258260, −1.29723956963573633539351893812, −0.901506178404924497312163161725, −0.21459013233408683789015183469,
0.21459013233408683789015183469, 0.901506178404924497312163161725, 1.29723956963573633539351893812, 1.64779610260291178492682258260, 2.03914371950139317304233224052, 2.40680611933508535784697927084, 2.41043263288142054287824514091, 3.29496070802326476320960186252, 3.30090896050323077553340695764, 3.63639334672869551973952218023, 3.80971575295479733257891132704, 3.88904228834475255816826712674, 3.98052925093805615692513134377, 4.33702238561373734486307242603, 4.78226975012131178775640686162, 5.11389822303637143725518061486, 5.35002193860007193946744069289, 5.40468767264128944277481186789, 5.62723625092631929489488712138, 5.94752062365395676645696621462, 6.16583355125133414259685367787, 6.26051792534973769158794296497, 6.93382898625983724569592332420, 6.94286987516285565539101555076, 7.10201365484174287963941947498
