L(s) = 1 | + 2·9-s − 11-s − 3·13-s − 4·16-s + 2·17-s − 6·19-s − 5·23-s − 6·25-s + 9·29-s − 14·31-s + 3·37-s − 5·41-s + 3·43-s − 3·47-s − 10·49-s − 5·53-s − 59-s − 9·61-s + 5·67-s − 71-s + 73-s + 4·79-s − 5·81-s + 25·83-s + 15·89-s + 8·97-s − 2·99-s + ⋯ |
L(s) = 1 | + 2/3·9-s − 0.301·11-s − 0.832·13-s − 16-s + 0.485·17-s − 1.37·19-s − 1.04·23-s − 6/5·25-s + 1.67·29-s − 2.51·31-s + 0.493·37-s − 0.780·41-s + 0.457·43-s − 0.437·47-s − 1.42·49-s − 0.686·53-s − 0.130·59-s − 1.15·61-s + 0.610·67-s − 0.118·71-s + 0.117·73-s + 0.450·79-s − 5/9·81-s + 2.74·83-s + 1.58·89-s + 0.812·97-s − 0.201·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100277 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100277 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 149 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 15 T + p T^{2} ) \) |
| 673 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 3 T - 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 115 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 48 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 5 T + 42 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T + 11 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 15 T + 188 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 6 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.3030475325, −13.7986985974, −13.4064840503, −12.9060928955, −12.5570236502, −12.1169932511, −11.7495978299, −11.1044707226, −10.6838747314, −10.3226983121, −9.70660404462, −9.45143118936, −8.92130387039, −8.18063269817, −7.88718255024, −7.39440000557, −6.75007466811, −6.34910746544, −5.79044547331, −4.95464493499, −4.63452670837, −3.96231625325, −3.31474041167, −2.25445272523, −1.83725972767, 0,
1.83725972767, 2.25445272523, 3.31474041167, 3.96231625325, 4.63452670837, 4.95464493499, 5.79044547331, 6.34910746544, 6.75007466811, 7.39440000557, 7.88718255024, 8.18063269817, 8.92130387039, 9.45143118936, 9.70660404462, 10.3226983121, 10.6838747314, 11.1044707226, 11.7495978299, 12.1169932511, 12.5570236502, 12.9060928955, 13.4064840503, 13.7986985974, 14.3030475325