| L(s) = 1 | + 2·2-s + 2·4-s − 4·7-s + 4·8-s − 8·14-s + 8·16-s + 4·23-s − 2·25-s − 8·28-s − 8·31-s + 8·32-s + 8·41-s + 8·46-s − 20·47-s − 12·49-s − 4·50-s − 16·56-s − 16·62-s + 8·64-s − 8·71-s + 16·73-s − 32·79-s + 16·82-s − 8·89-s + 8·92-s − 40·94-s − 16·97-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.51·7-s + 1.41·8-s − 2.13·14-s + 2·16-s + 0.834·23-s − 2/5·25-s − 1.51·28-s − 1.43·31-s + 1.41·32-s + 1.24·41-s + 1.17·46-s − 2.91·47-s − 1.71·49-s − 0.565·50-s − 2.13·56-s − 2.03·62-s + 64-s − 0.949·71-s + 1.87·73-s − 3.60·79-s + 1.76·82-s − 0.847·89-s + 0.834·92-s − 4.12·94-s − 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.233405385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.233405385\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_{4}$ | \( ( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 4266 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 3446 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 14294 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 11574 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 10506 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 308 T^{2} + 37386 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.399990160458342663903516350905, −7.939436505409647414691777318976, −7.63832181617016735843851816161, −7.51293825152653457695542570963, −7.26321586513784351464954705774, −6.90326086189370903788191931495, −6.74413697334034569185639019459, −6.42358565660020541450728758915, −6.38662618230933249468262056445, −5.72792070559792487855379137744, −5.66566895903323172678802855217, −5.62069900213332172794145655981, −5.20097136455862733667842748893, −4.65383738947388481465080438935, −4.55796050989934608835564951062, −4.39964039605826505730360080688, −4.12102533883116651466682002493, −3.44016511218030402097524064794, −3.26084289740073127740758956836, −3.14588198395061338889906045032, −3.11640880163608403230862313632, −2.19640443670185466365851002608, −1.83215813351008770961952748640, −1.58396509761193586148397804405, −0.56626102350494421880350944806,
0.56626102350494421880350944806, 1.58396509761193586148397804405, 1.83215813351008770961952748640, 2.19640443670185466365851002608, 3.11640880163608403230862313632, 3.14588198395061338889906045032, 3.26084289740073127740758956836, 3.44016511218030402097524064794, 4.12102533883116651466682002493, 4.39964039605826505730360080688, 4.55796050989934608835564951062, 4.65383738947388481465080438935, 5.20097136455862733667842748893, 5.62069900213332172794145655981, 5.66566895903323172678802855217, 5.72792070559792487855379137744, 6.38662618230933249468262056445, 6.42358565660020541450728758915, 6.74413697334034569185639019459, 6.90326086189370903788191931495, 7.26321586513784351464954705774, 7.51293825152653457695542570963, 7.63832181617016735843851816161, 7.939436505409647414691777318976, 8.399990160458342663903516350905
