L(s) = 1 | + 2-s + 2·3-s + 4-s − 2·5-s + 2·6-s − 3·7-s − 8-s − 2·10-s + 5·11-s + 2·12-s − 4·13-s − 3·14-s − 4·15-s − 3·16-s − 5·19-s − 2·20-s − 6·21-s + 5·22-s − 5·23-s − 2·24-s − 6·25-s − 4·26-s − 5·27-s − 3·28-s + 4·29-s − 4·30-s + 6·31-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.13·7-s − 0.353·8-s − 0.632·10-s + 1.50·11-s + 0.577·12-s − 1.10·13-s − 0.801·14-s − 1.03·15-s − 3/4·16-s − 1.14·19-s − 0.447·20-s − 1.30·21-s + 1.06·22-s − 1.04·23-s − 0.408·24-s − 6/5·25-s − 0.784·26-s − 0.962·27-s − 0.566·28-s + 0.742·29-s − 0.730·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125742 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125742 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - p T + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| 1103 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 23 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 9 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 56 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 44 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 79 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T - 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 11 T + 101 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 46 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 52 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−14.1117169025, −13.6752456111, −13.4188411097, −12.6016762172, −12.3334209707, −11.9967332612, −11.7459034528, −11.1995629532, −10.5217842505, −9.94588302812, −9.55156780198, −9.13125280364, −8.70794376098, −8.21644418721, −7.69837445974, −7.20536334131, −6.56788298689, −6.19671871418, −5.84839200751, −4.70455840057, −4.27879696993, −3.64719247933, −3.35609151934, −2.59780641028, −2.03646742978, 0,
2.03646742978, 2.59780641028, 3.35609151934, 3.64719247933, 4.27879696993, 4.70455840057, 5.84839200751, 6.19671871418, 6.56788298689, 7.20536334131, 7.69837445974, 8.21644418721, 8.70794376098, 9.13125280364, 9.55156780198, 9.94588302812, 10.5217842505, 11.1995629532, 11.7459034528, 11.9967332612, 12.3334209707, 12.6016762172, 13.4188411097, 13.6752456111, 14.1117169025