| L(s) = 1 | + 48·11-s − 120·17-s − 48·19-s − 164·25-s − 408·41-s − 528·43-s − 268·49-s + 1.87e3·59-s − 2.35e3·67-s + 968·73-s + 3.40e3·83-s + 3.67e3·89-s − 1.48e3·97-s + 2.83e3·107-s + 3.96e3·113-s − 2.86e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 148·169-s + 173-s + ⋯ |
| L(s) = 1 | + 1.31·11-s − 1.71·17-s − 0.579·19-s − 1.31·25-s − 1.55·41-s − 1.87·43-s − 0.781·49-s + 4.13·59-s − 4.28·67-s + 1.55·73-s + 4.50·83-s + 4.37·89-s − 1.54·97-s + 2.55·107-s + 3.29·113-s − 2.14·121-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 − 0.0673·169-s + 0.000439·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.106052634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.106052634\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 + 164 T^{2} + 19542 T^{4} + 164 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 268 T^{2} - 41658 T^{4} + 268 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 24 T + 2294 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 148 T^{2} + 3687126 T^{4} + 148 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 60 T + 6118 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 24 T + 9254 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 4900 T^{2} + 206522790 T^{4} - 4900 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 56516 T^{2} + 1986668214 T^{4} + 56516 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 63916 T^{2} + 2595558 p^{2} T^{4} + 63916 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 95476 T^{2} + 7028163510 T^{4} + 95476 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 204 T + 806 p T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 264 T + 171830 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 112892 T^{2} + 23215757766 T^{4} + 112892 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 507620 T^{2} + 107623720470 T^{4} + 507620 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 936 T + 498710 T^{2} - 936 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 132332 T^{2} + 107394800406 T^{4} - 132332 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 1176 T + 873542 T^{2} + 1176 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 4612 p T^{2} + 275267100390 T^{4} + 4612 p^{7} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 484 T + 541686 T^{2} - 484 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 932332 T^{2} + 435081520806 T^{4} + 932332 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 1704 T + 1721510 T^{2} - 1704 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 1836 T + 2086774 T^{2} - 1836 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 740 T + 1943814 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.16741131270362332517816639443, −5.85963954041651549585169120675, −5.72230119698521909991970881022, −5.25933769132499539673789025981, −5.21602697075368994009466817922, −4.80221867117595721619741711694, −4.77180111777568215562226885720, −4.73702556961456736166859354256, −4.34437302640048473488639577085, −4.00165861199247387868805113758, −3.78399194275148997442595816730, −3.71013249926740750311314974267, −3.61757608069927469560556811430, −3.13137155606989773665964087394, −3.12115674623909648131096333412, −2.56973628419862132670841545741, −2.48021710566931958501466072304, −1.90478737451995832263761888388, −1.88507311928098840215928641225, −1.76329445682349954455450751571, −1.72959040424689087341334889802, −0.848926869569398604333360920792, −0.818826500221001597908315377693, −0.46596083391684047030186343584, −0.28166791082850686229250937517,
0.28166791082850686229250937517, 0.46596083391684047030186343584, 0.818826500221001597908315377693, 0.848926869569398604333360920792, 1.72959040424689087341334889802, 1.76329445682349954455450751571, 1.88507311928098840215928641225, 1.90478737451995832263761888388, 2.48021710566931958501466072304, 2.56973628419862132670841545741, 3.12115674623909648131096333412, 3.13137155606989773665964087394, 3.61757608069927469560556811430, 3.71013249926740750311314974267, 3.78399194275148997442595816730, 4.00165861199247387868805113758, 4.34437302640048473488639577085, 4.73702556961456736166859354256, 4.77180111777568215562226885720, 4.80221867117595721619741711694, 5.21602697075368994009466817922, 5.25933769132499539673789025981, 5.72230119698521909991970881022, 5.85963954041651549585169120675, 6.16741131270362332517816639443
