L(s) = 1 | − 2-s − 2·3-s + 4-s + 5-s + 2·6-s − 3·8-s + 3·9-s − 10-s − 4·11-s − 2·12-s + 4·13-s − 2·15-s + 16-s + 11·17-s − 3·18-s − 7·19-s + 20-s + 4·22-s + 2·23-s + 6·24-s − 5·25-s − 4·26-s − 4·27-s − 10·29-s + 2·30-s + 31-s + 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 1.06·8-s + 9-s − 0.316·10-s − 1.20·11-s − 0.577·12-s + 1.10·13-s − 0.516·15-s + 1/4·16-s + 2.66·17-s − 0.707·18-s − 1.60·19-s + 0.223·20-s + 0.852·22-s + 0.417·23-s + 1.22·24-s − 25-s − 0.784·26-s − 0.769·27-s − 1.85·29-s + 0.365·30-s + 0.179·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11431161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11431161 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + p T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 7 T + 46 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 100 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 13 T + 90 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 56 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 20 T + 217 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 20 T + 229 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 15 T + 210 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 230 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.381063744404501940288256680448, −7.946733349766086775637059846075, −7.61249116716100040031737725899, −7.58052148896907937317999462687, −6.89890451855344137647384607653, −6.35314298476445282908764172026, −6.17308148295224066021950098869, −5.78201037381197237855648198410, −5.60418075489321244670031961377, −5.30823723209052916894712358424, −4.56145410477494174097432598502, −4.25362927966248744727776937245, −3.70997219923970979464632497441, −3.03476599192943406912289431201, −2.90466284649973946114324280253, −2.13260908374801624956106522384, −1.44187655184521798222135422784, −1.25892132810777911183587205575, 0, 0,
1.25892132810777911183587205575, 1.44187655184521798222135422784, 2.13260908374801624956106522384, 2.90466284649973946114324280253, 3.03476599192943406912289431201, 3.70997219923970979464632497441, 4.25362927966248744727776937245, 4.56145410477494174097432598502, 5.30823723209052916894712358424, 5.60418075489321244670031961377, 5.78201037381197237855648198410, 6.17308148295224066021950098869, 6.35314298476445282908764172026, 6.89890451855344137647384607653, 7.58052148896907937317999462687, 7.61249116716100040031737725899, 7.946733349766086775637059846075, 8.381063744404501940288256680448