| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s + 2·7-s − 4·8-s + 3·9-s − 4·10-s + 2·11-s − 6·12-s + 6·13-s − 4·14-s − 4·15-s + 5·16-s + 2·17-s − 6·18-s − 6·19-s + 6·20-s − 4·21-s − 4·22-s + 10·23-s + 8·24-s + 3·25-s − 12·26-s − 4·27-s + 6·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s − 1.26·10-s + 0.603·11-s − 1.73·12-s + 1.66·13-s − 1.06·14-s − 1.03·15-s + 5/4·16-s + 0.485·17-s − 1.41·18-s − 1.37·19-s + 1.34·20-s − 0.872·21-s − 0.852·22-s + 2.08·23-s + 1.63·24-s + 3/5·25-s − 2.35·26-s − 0.769·27-s + 1.13·28-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
| \(L(1)\) |
\(\approx\) |
\(2.101649749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.101649749\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 106 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_4$ | \( 1 - 22 T + 234 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 146 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 202 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.246304871298788178449959850323, −8.208534266707431486682652230463, −7.51318342509510167208885537849, −7.37052841116028662229569314900, −6.70560911867719498140269716049, −6.49439400923544669175429376317, −6.39595747005464852625255986944, −6.02151505197913308939441114342, −5.47391871651847778063827848429, −5.31774095714606819156059698923, −4.67449311775739098639510693736, −4.50818147697782063791262495360, −3.79173227099793657084379955812, −3.50229637168915671493615621073, −2.73123450546074790115526062579, −2.48007230798700502584987343320, −1.59130871902396746621714864172, −1.58371716344704851604750013521, −0.888091651674176792389231875468, −0.70430761721065147911808821057,
0.70430761721065147911808821057, 0.888091651674176792389231875468, 1.58371716344704851604750013521, 1.59130871902396746621714864172, 2.48007230798700502584987343320, 2.73123450546074790115526062579, 3.50229637168915671493615621073, 3.79173227099793657084379955812, 4.50818147697782063791262495360, 4.67449311775739098639510693736, 5.31774095714606819156059698923, 5.47391871651847778063827848429, 6.02151505197913308939441114342, 6.39595747005464852625255986944, 6.49439400923544669175429376317, 6.70560911867719498140269716049, 7.37052841116028662229569314900, 7.51318342509510167208885537849, 8.208534266707431486682652230463, 8.246304871298788178449959850323
