L(s) = 1 | − 18·9-s − 56·11-s + 12·19-s − 184·29-s − 244·31-s + 784·41-s + 426·49-s + 248·59-s + 1.50e3·61-s − 1.64e3·71-s − 1.76e3·79-s + 243·81-s + 1.72e3·89-s + 1.00e3·99-s + 2.01e3·101-s − 4.05e3·109-s − 2.75e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.10e3·169-s + ⋯ |
L(s) = 1 | − 2/3·9-s − 1.53·11-s + 0.144·19-s − 1.17·29-s − 1.41·31-s + 2.98·41-s + 1.24·49-s + 0.547·59-s + 3.14·61-s − 2.75·71-s − 2.50·79-s + 1/3·81-s + 2.05·89-s + 1.02·99-s + 1.98·101-s − 3.56·109-s − 2.07·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.501·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.225768508\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.225768508\) |
\(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $D_4\times C_2$ | \( 1 - 426 T^{2} + 75163 T^{4} - 426 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 28 T + 2554 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 1102 T^{2} + 8381283 T^{4} + 1102 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2140 T^{2} + 14277638 T^{4} - 2140 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 13423 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 34484 T^{2} + 545686278 T^{4} - 34484 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 92 T + 35998 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 122 T + 48407 T^{2} + 122 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 140396 T^{2} + 9176512758 T^{4} - 140396 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 392 T + 124882 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 79370 T^{2} + 14072890923 T^{4} - 79370 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 184180 T^{2} + 22560112358 T^{4} - 184180 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 206164 T^{2} + 44242847798 T^{4} - 206164 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 124 T + 336778 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 750 T + 535003 T^{2} - 750 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 758970 T^{2} + 300574469563 T^{4} - 758970 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 824 T + 877966 T^{2} + 824 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1547804 T^{2} + 901578574758 T^{4} - 1547804 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 880 T + 693278 T^{2} + 880 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 874148 T^{2} + 827869104438 T^{4} - 874148 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 864 T + 1202578 T^{2} - 864 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 1553905 T^{2} + p^{6} T^{4} )^{2} \) |
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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
−6.47793294598757004775987116402, −6.44774232635572910016293750434, −5.86897009901234364799008531158, −5.85256095244424261313719914745, −5.66942260432233953726970742188, −5.62459689528988808398976218116, −5.26065697335991506979567257351, −5.11118388542415773713891450225, −4.75953747566666428834963987920, −4.48862698021965612711391581300, −4.41729811100287958464120327110, −3.90318851727986959457799977599, −3.65255597212903549214591927310, −3.62215832498174715950786030892, −3.49606943700299413642403387292, −2.60953511303740975995487604555, −2.59967380986998907709268947022, −2.58677224690116237412795613278, −2.52036886696096643569050649872, −1.82892265725929209158952661518, −1.64402485410868897552531765326, −1.08469285403054432347843573352, −0.999963127928804309696004217790, −0.37233289806327344204094017498, −0.20268246308007988490693348643,
0.20268246308007988490693348643, 0.37233289806327344204094017498, 0.999963127928804309696004217790, 1.08469285403054432347843573352, 1.64402485410868897552531765326, 1.82892265725929209158952661518, 2.52036886696096643569050649872, 2.58677224690116237412795613278, 2.59967380986998907709268947022, 2.60953511303740975995487604555, 3.49606943700299413642403387292, 3.62215832498174715950786030892, 3.65255597212903549214591927310, 3.90318851727986959457799977599, 4.41729811100287958464120327110, 4.48862698021965612711391581300, 4.75953747566666428834963987920, 5.11118388542415773713891450225, 5.26065697335991506979567257351, 5.62459689528988808398976218116, 5.66942260432233953726970742188, 5.85256095244424261313719914745, 5.86897009901234364799008531158, 6.44774232635572910016293750434, 6.47793294598757004775987116402