| L(s) = 1 | − 8·13-s + 32·19-s − 8·31-s − 24·37-s + 144·43-s − 176·49-s + 152·61-s − 16·67-s − 152·73-s + 376·79-s − 120·97-s + 544·103-s − 472·109-s + 440·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 616·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 0.615·13-s + 1.68·19-s − 0.258·31-s − 0.648·37-s + 3.34·43-s − 3.59·49-s + 2.49·61-s − 0.238·67-s − 2.08·73-s + 4.75·79-s − 1.23·97-s + 5.28·103-s − 4.33·109-s + 3.63·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 3.64·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(9.726728278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.726728278\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 88 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 40 p T^{2} + 77522 T^{4} - 40 p^{5} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 332 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 258 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 16 T + 426 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1740 T^{2} + 1310822 T^{4} - 1740 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1148 T^{2} + 1242278 T^{4} - 1148 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 926 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 12 T + 2684 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 4160 T^{2} + 9967682 T^{4} - 4160 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 72 T + 3034 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8532 T^{2} + 27935078 T^{4} - 8532 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4052 T^{2} + 8048198 T^{4} - 4052 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8120 T^{2} + 33184082 T^{4} - 8120 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 76 T + 6926 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 8 T + 3234 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 13540 T^{2} + 89191302 T^{4} - 13540 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 76 T + 7262 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 188 T + 20318 T^{2} - 188 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 17300 T^{2} + 169575302 T^{4} - 17300 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 19424 T^{2} + 217871426 T^{4} - 19424 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 60 T + 14878 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−5.97682716185840717387059752639, −5.54101820044735269267998441835, −5.35009336184170470668632494250, −5.29272558510700140765114121667, −5.25392904694368649546177930242, −4.81889779303084085521307075087, −4.65415951230742306529579996596, −4.51131504707506092248917400577, −4.34950273371839557916651096416, −3.83786404989507840388412517687, −3.82777618213221128747137559520, −3.76420890044119527885639648084, −3.28462460216897204853000452217, −2.98375206426856917185537235390, −2.97797340934003898942848594598, −2.92787003581686632633143134625, −2.46886755687524188603183478788, −1.98523290409100397251199286106, −1.98394698743368914461751695975, −1.79396046282959351882794502545, −1.54005646830467548797453154301, −0.865294842107783622819918906311, −0.850099442435875600340150643834, −0.50375512634117618409836865628, −0.44009444796070277031978248387,
0.44009444796070277031978248387, 0.50375512634117618409836865628, 0.850099442435875600340150643834, 0.865294842107783622819918906311, 1.54005646830467548797453154301, 1.79396046282959351882794502545, 1.98394698743368914461751695975, 1.98523290409100397251199286106, 2.46886755687524188603183478788, 2.92787003581686632633143134625, 2.97797340934003898942848594598, 2.98375206426856917185537235390, 3.28462460216897204853000452217, 3.76420890044119527885639648084, 3.82777618213221128747137559520, 3.83786404989507840388412517687, 4.34950273371839557916651096416, 4.51131504707506092248917400577, 4.65415951230742306529579996596, 4.81889779303084085521307075087, 5.25392904694368649546177930242, 5.29272558510700140765114121667, 5.35009336184170470668632494250, 5.54101820044735269267998441835, 5.97682716185840717387059752639
