| L(s) = 1 | + 32·7-s + 486·11-s + 208·13-s − 1.94e3·17-s + 632·19-s + 6.80e3·23-s − 2.92e3·25-s − 1.16e4·29-s − 3.32e3·31-s − 9.95e3·37-s − 1.36e4·41-s − 4.96e3·43-s + 1.84e4·47-s − 2.95e3·49-s + 1.16e4·53-s + 1.94e3·59-s + 8.17e3·61-s − 9.00e4·67-s − 4.47e4·71-s − 1.21e5·73-s + 1.55e4·77-s − 2.87e4·79-s − 1.50e4·83-s + 1.78e5·89-s + 6.65e3·91-s + 8.89e4·97-s + 2.02e5·101-s + ⋯ |
| L(s) = 1 | + 0.246·7-s + 1.21·11-s + 0.341·13-s − 1.63·17-s + 0.401·19-s + 2.68·23-s − 0.937·25-s − 2.57·29-s − 0.621·31-s − 1.19·37-s − 1.26·41-s − 0.409·43-s + 1.21·47-s − 0.175·49-s + 0.570·53-s + 0.0727·59-s + 0.281·61-s − 2.45·67-s − 1.05·71-s − 2.66·73-s + 0.298·77-s − 0.518·79-s − 0.240·83-s + 2.39·89-s + 0.0842·91-s + 0.959·97-s + 1.97·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.309206255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.309206255\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2929 T^{2} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 32 T + 3981 T^{2} - 32 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 486 T + 168607 T^{2} - 486 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 16 p T + 275178 T^{2} - 16 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1944 T + 2456098 T^{2} + 1944 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 632 T + 4932498 T^{2} - 632 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6804 T + 24393154 T^{2} - 6804 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 11664 T + 70238998 T^{2} + 11664 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3328 T + 38238117 T^{2} + 3328 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 9956 T + 151512798 T^{2} + 9956 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 13608 T + 266834974 T^{2} + 13608 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4960 T + 213010962 T^{2} + 4960 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18468 T + 312496354 T^{2} - 18468 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11664 T + 484234009 T^{2} - 11664 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1944 T + 961283686 T^{2} - 1944 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8176 T - 15702054 T^{2} - 8176 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 90064 T + 4055628738 T^{2} + 90064 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 44712 T + 2046094414 T^{2} + 44712 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 121214 T + 6975764499 T^{2} + 121214 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 28768 T + 4017714654 T^{2} + 28768 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 15066 T - 2258995409 T^{2} + 15066 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 178848 T + 18739138030 T^{2} - 178848 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 88942 T + 13779025491 T^{2} - 88942 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.43215611654200321509609971522, −10.41093307072862218884768648862, −9.337689473667985481524753281947, −9.208821498389723535452953158741, −8.797943111336598598011869727430, −8.646537785530232555318159259718, −7.59123864364666994351245313398, −7.30596286425740586848844083994, −6.94495514174418941698002360978, −6.47054693249761308194132608641, −5.64714001133398423275566134476, −5.55270192527791232914573088848, −4.59772058050846112738465323454, −4.39662967964700743164105523999, −3.42921592771555868697544521225, −3.40814639383635795985366151710, −2.32005564578179229863807587797, −1.66650341883960514357445063835, −1.30159302448032663499812578669, −0.27498554644641973937916250324,
0.27498554644641973937916250324, 1.30159302448032663499812578669, 1.66650341883960514357445063835, 2.32005564578179229863807587797, 3.40814639383635795985366151710, 3.42921592771555868697544521225, 4.39662967964700743164105523999, 4.59772058050846112738465323454, 5.55270192527791232914573088848, 5.64714001133398423275566134476, 6.47054693249761308194132608641, 6.94495514174418941698002360978, 7.30596286425740586848844083994, 7.59123864364666994351245313398, 8.646537785530232555318159259718, 8.797943111336598598011869727430, 9.208821498389723535452953158741, 9.337689473667985481524753281947, 10.41093307072862218884768648862, 10.43215611654200321509609971522
