| L(s) = 1 | + 4·2-s + 8·4-s + 4·5-s + 6·7-s + 8·8-s + 16·10-s − 6·13-s + 24·14-s − 4·16-s + 32·20-s + 11·25-s − 24·26-s + 48·28-s − 32·32-s + 24·35-s + 6·37-s + 32·40-s − 16·47-s + 13·49-s + 44·50-s − 48·52-s + 48·56-s − 6·61-s − 64·64-s − 24·65-s − 24·67-s + 96·70-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s + 1.78·5-s + 2.26·7-s + 2.82·8-s + 5.05·10-s − 1.66·13-s + 6.41·14-s − 16-s + 7.15·20-s + 11/5·25-s − 4.70·26-s + 9.07·28-s − 5.65·32-s + 4.05·35-s + 0.986·37-s + 5.05·40-s − 2.33·47-s + 13/7·49-s + 6.22·50-s − 6.65·52-s + 6.41·56-s − 0.768·61-s − 8·64-s − 2.97·65-s − 2.93·67-s + 11.4·70-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.27506342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.27506342\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 177 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.97844215882408177988573474519, −10.93920089835723928221966530646, −10.14089860039039578414241450246, −9.704887484850753901636210390500, −9.194915851832507433526147382543, −8.856831722026921941714036657255, −8.045242613288293194955200531335, −7.72419418467460300796419528806, −6.94627373863877983068362298647, −6.61900283515020463328930676205, −5.98287322860284498591810825816, −5.66842890621948180439326955837, −5.16056740807084470282749955218, −4.88271407134164340871263453969, −4.59390406039760231689655375873, −4.21242420726856230888366215458, −3.02844831692419744500769815553, −2.85497385576384006808620357979, −1.90306447948722643718359930552, −1.81519697457318625967382379853,
1.81519697457318625967382379853, 1.90306447948722643718359930552, 2.85497385576384006808620357979, 3.02844831692419744500769815553, 4.21242420726856230888366215458, 4.59390406039760231689655375873, 4.88271407134164340871263453969, 5.16056740807084470282749955218, 5.66842890621948180439326955837, 5.98287322860284498591810825816, 6.61900283515020463328930676205, 6.94627373863877983068362298647, 7.72419418467460300796419528806, 8.045242613288293194955200531335, 8.856831722026921941714036657255, 9.194915851832507433526147382543, 9.704887484850753901636210390500, 10.14089860039039578414241450246, 10.93920089835723928221966530646, 10.97844215882408177988573474519
