| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 2·5-s + 4·6-s + 7-s + 4·8-s + 3·9-s + 4·10-s + 2·11-s + 6·12-s − 2·13-s + 2·14-s + 4·15-s + 5·16-s + 2·17-s + 6·18-s + 8·19-s + 6·20-s + 2·21-s + 4·22-s − 23-s + 8·24-s + 3·25-s − 4·26-s + 4·27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 0.377·7-s + 1.41·8-s + 9-s + 1.26·10-s + 0.603·11-s + 1.73·12-s − 0.554·13-s + 0.534·14-s + 1.03·15-s + 5/4·16-s + 0.485·17-s + 1.41·18-s + 1.83·19-s + 1.34·20-s + 0.436·21-s + 0.852·22-s − 0.208·23-s + 1.63·24-s + 3/5·25-s − 0.784·26-s + 0.769·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(22.86523764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.86523764\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + T + 38 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 80 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 15 T + 110 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 9 T + 98 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T - 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 158 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 22 T + 266 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 228 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−8.299900316175089647572466110095, −8.069280143803394850199222716298, −7.42812289881646403344433528819, −7.07765490772405268585910399991, −6.90920392381844580148155913872, −6.63291035737035358774974795939, −5.96931022098418547751769758261, −5.92568207832459889250767163038, −5.13242157497669802015675125328, −5.10628287984761917409388676712, −4.58237972667168940730445533102, −4.53777329724031084095520610088, −3.62415153635194243712689392677, −3.52177513816049189063311084093, −2.99486759739824171108882606540, −2.82908830048746909481524128672, −2.30008429836263018732977657863, −1.75888445558378289494918619386, −1.37188194691573686285945082571, −0.929298977080635806552515375300,
0.929298977080635806552515375300, 1.37188194691573686285945082571, 1.75888445558378289494918619386, 2.30008429836263018732977657863, 2.82908830048746909481524128672, 2.99486759739824171108882606540, 3.52177513816049189063311084093, 3.62415153635194243712689392677, 4.53777329724031084095520610088, 4.58237972667168940730445533102, 5.10628287984761917409388676712, 5.13242157497669802015675125328, 5.92568207832459889250767163038, 5.96931022098418547751769758261, 6.63291035737035358774974795939, 6.90920392381844580148155913872, 7.07765490772405268585910399991, 7.42812289881646403344433528819, 8.069280143803394850199222716298, 8.299900316175089647572466110095
