| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 2·5-s + 4·6-s + 4·7-s + 4·8-s + 3·9-s + 4·10-s + 6·12-s + 6·13-s + 8·14-s + 4·15-s + 5·16-s + 4·17-s + 6·18-s − 2·19-s + 6·20-s + 8·21-s − 6·23-s + 8·24-s + 3·25-s + 12·26-s + 4·27-s + 12·28-s − 6·29-s + 8·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 1.51·7-s + 1.41·8-s + 9-s + 1.26·10-s + 1.73·12-s + 1.66·13-s + 2.13·14-s + 1.03·15-s + 5/4·16-s + 0.970·17-s + 1.41·18-s − 0.458·19-s + 1.34·20-s + 1.74·21-s − 1.25·23-s + 1.63·24-s + 3/5·25-s + 2.35·26-s + 0.769·27-s + 2.26·28-s − 1.11·29-s + 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(22.83562022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.83562022\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 108 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 14 T + 152 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 92 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 104 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 120 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 59 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 34 T + 480 T^{2} + 34 p T^{3} + p^{2} T^{4} \) |
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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
−8.394268099470493082402590767464, −8.387457645867617326135688056787, −7.910554230765527998978303499899, −7.81703851237210618467699636495, −7.07747027785791675460445911131, −7.02546373954685779334712628009, −6.17863995950606854690903728613, −6.09156048186245265312117252397, −5.60901656647989014352336919055, −5.54518407433964995818255149902, −4.73722295130883785821621129594, −4.40148008582044716569074615489, −4.23729570019971350028836290756, −3.73239373445252563354893589814, −3.18675843951102880262587063785, −2.94969540625736663893497656088, −2.13428281247480610256209125136, −2.08239217263590813845521553259, −1.33019107023578876903192351829, −1.21369092461611589623152014380,
1.21369092461611589623152014380, 1.33019107023578876903192351829, 2.08239217263590813845521553259, 2.13428281247480610256209125136, 2.94969540625736663893497656088, 3.18675843951102880262587063785, 3.73239373445252563354893589814, 4.23729570019971350028836290756, 4.40148008582044716569074615489, 4.73722295130883785821621129594, 5.54518407433964995818255149902, 5.60901656647989014352336919055, 6.09156048186245265312117252397, 6.17863995950606854690903728613, 7.02546373954685779334712628009, 7.07747027785791675460445911131, 7.81703851237210618467699636495, 7.910554230765527998978303499899, 8.387457645867617326135688056787, 8.394268099470493082402590767464
