| L(s) = 1 | − 9·9-s + 8·11-s − 88·19-s − 268·29-s − 544·31-s − 12·41-s + 686·49-s − 72·59-s − 884·61-s − 1.26e3·71-s − 1.37e3·79-s + 81·81-s + 1.38e3·89-s − 72·99-s + 2.36e3·101-s − 684·109-s − 2.61e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.47e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 0.219·11-s − 1.06·19-s − 1.71·29-s − 3.15·31-s − 0.0457·41-s + 2·49-s − 0.158·59-s − 1.85·61-s − 2.11·71-s − 1.95·79-s + 1/9·81-s + 1.65·89-s − 0.0730·99-s + 2.32·101-s − 0.601·109-s − 1.96·121-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 + 0.672·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3179118578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3179118578\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1478 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3170 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 44 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 15118 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 134 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 272 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 91702 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 158870 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 167646 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 129962 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 442 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 566182 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 632 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 625934 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 688 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 267770 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 694 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1153730 T^{2} + p^{6} T^{4} \) |
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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
−10.69629541549942725807611682851, −10.25633886596405723429237417481, −9.332510511849555237361469296448, −9.054182899510504767428200767745, −9.024341428141755087438919507837, −8.376890153181329155951615592856, −7.63250972582140322417083208244, −7.39556016433475882080620057754, −7.12510404396084307211466742553, −6.14613289352257363370544178435, −6.10434787250119803042642376667, −5.42369418229241331812313296898, −5.06780187760752822596042140395, −4.20201839373411815989632100324, −3.90393269393095873265009837721, −3.35542782047654694547117240842, −2.57770902068660071515558410836, −1.93134689544943063051462681477, −1.41484590895663596419231647302, −0.16056113910555679781923127243,
0.16056113910555679781923127243, 1.41484590895663596419231647302, 1.93134689544943063051462681477, 2.57770902068660071515558410836, 3.35542782047654694547117240842, 3.90393269393095873265009837721, 4.20201839373411815989632100324, 5.06780187760752822596042140395, 5.42369418229241331812313296898, 6.10434787250119803042642376667, 6.14613289352257363370544178435, 7.12510404396084307211466742553, 7.39556016433475882080620057754, 7.63250972582140322417083208244, 8.376890153181329155951615592856, 9.024341428141755087438919507837, 9.054182899510504767428200767745, 9.332510511849555237361469296448, 10.25633886596405723429237417481, 10.69629541549942725807611682851
