| L(s) = 1 | + 2·3-s + 2·5-s − 9-s + 2·11-s − 10·13-s + 4·15-s − 10·17-s − 4·19-s − 4·23-s + 3·25-s − 6·27-s + 2·29-s + 12·31-s + 4·33-s − 4·37-s − 20·39-s − 4·41-s − 12·43-s − 2·45-s + 2·47-s − 20·51-s − 16·53-s + 4·55-s − 8·57-s + 4·59-s − 16·61-s − 20·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 1/3·9-s + 0.603·11-s − 2.77·13-s + 1.03·15-s − 2.42·17-s − 0.917·19-s − 0.834·23-s + 3/5·25-s − 1.15·27-s + 0.371·29-s + 2.15·31-s + 0.696·33-s − 0.657·37-s − 3.20·39-s − 0.624·41-s − 1.82·43-s − 0.298·45-s + 0.291·47-s − 2.80·51-s − 2.19·53-s + 0.539·55-s − 1.05·57-s + 0.520·59-s − 2.04·61-s − 2.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10 T + 49 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 96 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 152 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 100 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T - 41 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.411549592356559750202384864919, −8.057624334172226521205324352234, −7.63194078221399443906328334554, −7.12027456234924130525223839857, −6.75170913865931089181132474826, −6.62641694810354889980586563675, −5.97784003847560697057061298479, −5.97720642831026896358276048095, −5.01300599103088654879026902206, −4.83239740943215914542477096164, −4.50247830156430512006234874105, −4.30834083222437370162118959945, −3.28828092338061938162100098841, −3.20011657130695942792422238083, −2.51549787270007322302359770170, −2.27507569614308910622607640096, −2.08814180239438050294346834022, −1.44195824761893878378737535022, 0, 0,
1.44195824761893878378737535022, 2.08814180239438050294346834022, 2.27507569614308910622607640096, 2.51549787270007322302359770170, 3.20011657130695942792422238083, 3.28828092338061938162100098841, 4.30834083222437370162118959945, 4.50247830156430512006234874105, 4.83239740943215914542477096164, 5.01300599103088654879026902206, 5.97720642831026896358276048095, 5.97784003847560697057061298479, 6.62641694810354889980586563675, 6.75170913865931089181132474826, 7.12027456234924130525223839857, 7.63194078221399443906328334554, 8.057624334172226521205324352234, 8.411549592356559750202384864919
