| L(s) = 1 | + 2·3-s − 2·4-s − 3·7-s − 11-s − 4·12-s − 13-s − 5·19-s − 6·21-s + 2·23-s − 4·25-s − 2·27-s + 6·28-s + 29-s − 2·31-s − 2·33-s + 3·37-s − 2·39-s − 6·41-s + 43-s + 2·44-s − 3·47-s − 6·49-s + 2·52-s − 3·53-s − 10·57-s − 9·59-s + 3·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 1.13·7-s − 0.301·11-s − 1.15·12-s − 0.277·13-s − 1.14·19-s − 1.30·21-s + 0.417·23-s − 4/5·25-s − 0.384·27-s + 1.13·28-s + 0.185·29-s − 0.359·31-s − 0.348·33-s + 0.493·37-s − 0.320·39-s − 0.937·41-s + 0.152·43-s + 0.301·44-s − 0.437·47-s − 6/7·49-s + 0.277·52-s − 0.412·53-s − 1.32·57-s − 1.17·59-s + 0.384·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46315 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46315 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 10 T + p T^{2} ) \) |
| 157 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 24 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 10 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 12 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 64 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T - 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 59 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T - 69 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T - 106 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 174 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−14.8996531292, −14.5879260415, −14.0465134544, −13.7385200338, −13.2758321090, −12.8927126844, −12.6477888340, −11.9616124700, −11.3363252464, −10.8010774856, −10.1262360124, −9.77035118581, −9.29018424261, −8.91626045195, −8.48692166684, −7.98943614039, −7.49333352906, −6.55634912169, −6.35870444720, −5.42616173030, −4.79224556290, −4.13307858976, −3.43165893099, −2.91487481554, −2.07164966649, 0,
2.07164966649, 2.91487481554, 3.43165893099, 4.13307858976, 4.79224556290, 5.42616173030, 6.35870444720, 6.55634912169, 7.49333352906, 7.98943614039, 8.48692166684, 8.91626045195, 9.29018424261, 9.77035118581, 10.1262360124, 10.8010774856, 11.3363252464, 11.9616124700, 12.6477888340, 12.8927126844, 13.2758321090, 13.7385200338, 14.0465134544, 14.5879260415, 14.8996531292
