L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 2·5-s + 4·6-s − 2·7-s + 4·8-s + 2·9-s + 4·10-s − 5·11-s + 4·12-s − 3·13-s − 4·14-s + 4·15-s + 8·16-s − 2·17-s + 4·18-s + 4·19-s + 4·20-s − 4·21-s − 10·22-s + 5·23-s + 8·24-s + 2·25-s − 6·26-s + 6·27-s − 4·28-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 0.894·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s + 2/3·9-s + 1.26·10-s − 1.50·11-s + 1.15·12-s − 0.832·13-s − 1.06·14-s + 1.03·15-s + 2·16-s − 0.485·17-s + 0.942·18-s + 0.917·19-s + 0.894·20-s − 0.872·21-s − 2.13·22-s + 1.04·23-s + 1.63·24-s + 2/5·25-s − 1.17·26-s + 1.15·27-s − 0.755·28-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
\(L(1)\) |
\(\approx\) |
\(5.063458818\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.063458818\) |
\(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T )( 1 - 9 T + p T^{2} ) \) |
| 673 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 41 T + p T^{2} ) \) |
good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 36 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 9 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 56 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T - 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 68 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 80 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 59 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T - 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 17 T + 170 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T + 58 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 160 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−13.9474051464, −13.5312582377, −13.3350792872, −13.0167716432, −12.5933001606, −12.2380024832, −11.4964198713, −10.8822128938, −10.3985108225, −10.1400538147, −9.50339092291, −9.32205457854, −8.39352811093, −8.11732528511, −7.43992627883, −7.03057676304, −6.58490574830, −5.64718744355, −5.39225260525, −4.76566931995, −4.37622840695, −3.42544924403, −2.80656417728, −2.65398446953, −1.57749177230,
1.57749177230, 2.65398446953, 2.80656417728, 3.42544924403, 4.37622840695, 4.76566931995, 5.39225260525, 5.64718744355, 6.58490574830, 7.03057676304, 7.43992627883, 8.11732528511, 8.39352811093, 9.32205457854, 9.50339092291, 10.1400538147, 10.3985108225, 10.8822128938, 11.4964198713, 12.2380024832, 12.5933001606, 13.0167716432, 13.3350792872, 13.5312582377, 13.9474051464