| L(s) = 1 | − 2·7-s + 6·9-s − 8·16-s + 16·19-s + 14·31-s + 20·37-s + 2·49-s − 12·63-s + 22·67-s + 34·73-s + 27·81-s − 10·97-s − 38·109-s + 16·112-s + 127-s + 131-s − 32·133-s + 137-s + 139-s − 48·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 96·171-s + 173-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 2·9-s − 2·16-s + 3.67·19-s + 2.51·31-s + 3.28·37-s + 2/7·49-s − 1.51·63-s + 2.68·67-s + 3.97·73-s + 3·81-s − 1.01·97-s − 3.63·109-s + 1.51·112-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-s + 0.0854·137-s + 0.0848·139-s − 4·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 7.34·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.659169431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.659169431\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 47 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 - 169 T^{2} + p^{2} T^{4} ) \) |
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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
−7.982944260701465617444260701789, −7.64052961419068532193081485968, −7.51258809412948620450992386380, −6.93715516477283072681580346185, −6.84813306304365579723562258429, −6.78623665906618647285288540092, −6.57659810474222086040522359080, −6.34262071818743938433338045813, −5.97955408574736551149295518241, −5.57788630298500011433635050560, −5.29947676156940474400782447834, −5.17541792611726526999647179595, −4.81102149765412950041136046073, −4.50450951165875432678108561816, −4.40671778278207042767954334882, −3.99630606893832862879072928546, −3.79870781460858620263530819883, −3.37122348620860541365737243198, −3.16721810704272953331780550100, −2.61928471810986837041310469127, −2.36098811863784298264928976605, −2.31033352261935587134182773565, −1.37712763592021598865447856777, −0.946722817085569674300530009746, −0.894111108031332083036666042619,
0.894111108031332083036666042619, 0.946722817085569674300530009746, 1.37712763592021598865447856777, 2.31033352261935587134182773565, 2.36098811863784298264928976605, 2.61928471810986837041310469127, 3.16721810704272953331780550100, 3.37122348620860541365737243198, 3.79870781460858620263530819883, 3.99630606893832862879072928546, 4.40671778278207042767954334882, 4.50450951165875432678108561816, 4.81102149765412950041136046073, 5.17541792611726526999647179595, 5.29947676156940474400782447834, 5.57788630298500011433635050560, 5.97955408574736551149295518241, 6.34262071818743938433338045813, 6.57659810474222086040522359080, 6.78623665906618647285288540092, 6.84813306304365579723562258429, 6.93715516477283072681580346185, 7.51258809412948620450992386380, 7.64052961419068532193081485968, 7.982944260701465617444260701789
