| L(s) = 1 | + 2·4-s − 2·7-s + 9-s − 4·13-s + 16-s − 25-s − 4·28-s + 2·31-s + 2·36-s + 4·37-s − 2·43-s + 49-s − 8·52-s − 2·63-s − 2·64-s − 2·73-s − 2·79-s + 8·91-s + 2·97-s − 2·100-s − 2·103-s − 2·109-s − 2·112-s − 4·117-s − 2·121-s + 4·124-s + 127-s + ⋯ |
| L(s) = 1 | + 2·4-s − 2·7-s + 9-s − 4·13-s + 16-s − 25-s − 4·28-s + 2·31-s + 2·36-s + 4·37-s − 2·43-s + 49-s − 8·52-s − 2·63-s − 2·64-s − 2·73-s − 2·79-s + 8·91-s + 2·97-s − 2·100-s − 2·103-s − 2·109-s − 2·112-s − 4·117-s − 2·121-s + 4·124-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3821155279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3821155279\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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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
−9.261984649232099377938399527062, −8.657188688802703417166928955622, −8.231170295604727435905978789256, −8.007607814910001464625463047681, −7.71822027925855096827287524729, −7.51828293431547257653370785905, −7.22864699467596249750084034109, −7.16237759234306319121707143661, −6.79630155492941171133576287977, −6.72194297379546005470977822708, −6.29154362102306211253296752947, −6.11384775433892806220920232724, −6.08316196652020099419979268741, −5.48172561341015488566674130080, −5.07790150804553031667076782138, −4.79555038552527192406901869376, −4.49079745162150097735768487124, −4.21133916261879778572438295852, −3.99770012676609839692866788023, −3.08829971249185133363349303392, −2.92245919067083545283945155759, −2.75991142853910062421931303486, −2.52377090036080620067152435299, −2.09368771117900151686760285036, −1.56870668635183199225822516103,
1.56870668635183199225822516103, 2.09368771117900151686760285036, 2.52377090036080620067152435299, 2.75991142853910062421931303486, 2.92245919067083545283945155759, 3.08829971249185133363349303392, 3.99770012676609839692866788023, 4.21133916261879778572438295852, 4.49079745162150097735768487124, 4.79555038552527192406901869376, 5.07790150804553031667076782138, 5.48172561341015488566674130080, 6.08316196652020099419979268741, 6.11384775433892806220920232724, 6.29154362102306211253296752947, 6.72194297379546005470977822708, 6.79630155492941171133576287977, 7.16237759234306319121707143661, 7.22864699467596249750084034109, 7.51828293431547257653370785905, 7.71822027925855096827287524729, 8.007607814910001464625463047681, 8.231170295604727435905978789256, 8.657188688802703417166928955622, 9.261984649232099377938399527062
