| L(s) = 1 | − 2·2-s + 3·4-s + 2·7-s − 6·8-s − 9-s − 4·14-s + 9·16-s + 2·18-s + 2·19-s − 4·25-s + 6·28-s − 12·32-s − 3·36-s − 4·38-s + 2·41-s + 3·49-s + 8·50-s − 12·56-s + 2·59-s − 2·63-s + 18·64-s + 2·67-s − 2·71-s + 6·72-s + 6·76-s + 81-s − 4·82-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s + 2·7-s − 6·8-s − 9-s − 4·14-s + 9·16-s + 2·18-s + 2·19-s − 4·25-s + 6·28-s − 12·32-s − 3·36-s − 4·38-s + 2·41-s + 3·49-s + 8·50-s − 12·56-s + 2·59-s − 2·63-s + 18·64-s + 2·67-s − 2·71-s + 6·72-s + 6·76-s + 81-s − 4·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1889996987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1889996987\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 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
−8.419997929838757551138900962459, −8.216775647198376174303104780645, −7.88055998241061986020108531661, −7.80136424471324208933566855871, −7.54937358194585242573163224180, −7.50834843972665177469714238890, −6.99341365948518465897903539028, −6.80157891457166250840806925404, −6.34264288123609241439954148522, −6.19859338694373332161603244505, −5.98085371872081457354262675499, −5.53310492123583210019487167153, −5.35443994116059322496251342488, −5.34181723325164253808521418908, −5.32693212338431208144727800978, −4.26209016594983671575054917896, −3.94653302738206801344771665441, −3.87440947248701690403815744092, −3.48762579600775443326674568583, −2.82270671716034486608668257237, −2.66549256429074363496068972055, −2.53886931670367344245009654332, −1.97586661485529291718951358492, −1.60641294567980630786349282588, −0.977486845819645189753550122434,
0.977486845819645189753550122434, 1.60641294567980630786349282588, 1.97586661485529291718951358492, 2.53886931670367344245009654332, 2.66549256429074363496068972055, 2.82270671716034486608668257237, 3.48762579600775443326674568583, 3.87440947248701690403815744092, 3.94653302738206801344771665441, 4.26209016594983671575054917896, 5.32693212338431208144727800978, 5.34181723325164253808521418908, 5.35443994116059322496251342488, 5.53310492123583210019487167153, 5.98085371872081457354262675499, 6.19859338694373332161603244505, 6.34264288123609241439954148522, 6.80157891457166250840806925404, 6.99341365948518465897903539028, 7.50834843972665177469714238890, 7.54937358194585242573163224180, 7.80136424471324208933566855871, 7.88055998241061986020108531661, 8.216775647198376174303104780645, 8.419997929838757551138900962459
