| L(s) = 1 | − 2·3-s − 2·5-s + 7-s + 3·9-s + 2·11-s + 4·13-s + 4·15-s + 5·17-s + 6·19-s − 2·21-s + 2·23-s + 3·25-s − 4·27-s + 13·29-s − 5·31-s − 4·33-s − 2·35-s − 3·37-s − 8·39-s + 11·41-s − 4·43-s − 6·45-s − 16·47-s − 9·49-s − 10·51-s − 3·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.377·7-s + 9-s + 0.603·11-s + 1.10·13-s + 1.03·15-s + 1.21·17-s + 1.37·19-s − 0.436·21-s + 0.417·23-s + 3/5·25-s − 0.769·27-s + 2.41·29-s − 0.898·31-s − 0.696·33-s − 0.338·35-s − 0.493·37-s − 1.28·39-s + 1.71·41-s − 0.609·43-s − 0.894·45-s − 2.33·47-s − 9/7·49-s − 1.40·51-s − 0.412·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.207551542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.207551542\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 72 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 11 T + 108 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 104 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 9 T + 134 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 130 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 134 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 13 T + 170 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.215450934139831837003691124136, −7.940818418892383578697127069368, −7.50190400208261241211237728138, −7.43041125398753764532457731840, −6.70084149423345192745244626318, −6.64000514626343515321243053579, −6.09816665520392107885204761181, −6.00464279364606077396879937158, −5.22099792267254017349961293861, −5.21321340797861564624400875713, −4.69798395726745446966959343306, −4.44386113429387567014438307518, −3.91471208576146672670241265459, −3.45924325758709202291374021424, −3.13413402679821887791167082729, −2.90298709815738285260474802362, −1.73486531930053863556118194325, −1.50309366246724326555760133871, −0.942570827195454114134454540525, −0.55288318881786240260948979904,
0.55288318881786240260948979904, 0.942570827195454114134454540525, 1.50309366246724326555760133871, 1.73486531930053863556118194325, 2.90298709815738285260474802362, 3.13413402679821887791167082729, 3.45924325758709202291374021424, 3.91471208576146672670241265459, 4.44386113429387567014438307518, 4.69798395726745446966959343306, 5.21321340797861564624400875713, 5.22099792267254017349961293861, 6.00464279364606077396879937158, 6.09816665520392107885204761181, 6.64000514626343515321243053579, 6.70084149423345192745244626318, 7.43041125398753764532457731840, 7.50190400208261241211237728138, 7.940818418892383578697127069368, 8.215450934139831837003691124136
