| L(s) = 1 | − 2·2-s − 4·3-s + 4-s − 5·5-s + 8·6-s − 6·7-s + 2·8-s + 7·9-s + 10·10-s − 2·11-s − 4·12-s − 7·13-s + 12·14-s + 20·15-s − 3·16-s − 10·17-s − 14·18-s − 3·19-s − 5·20-s + 24·21-s + 4·22-s − 23-s − 8·24-s + 12·25-s + 14·26-s − 4·27-s − 6·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 1/2·4-s − 2.23·5-s + 3.26·6-s − 2.26·7-s + 0.707·8-s + 7/3·9-s + 3.16·10-s − 0.603·11-s − 1.15·12-s − 1.94·13-s + 3.20·14-s + 5.16·15-s − 3/4·16-s − 2.42·17-s − 3.29·18-s − 0.688·19-s − 1.11·20-s + 5.23·21-s + 0.852·22-s − 0.208·23-s − 1.63·24-s + 12/5·25-s + 2.74·26-s − 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52498 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52498 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 26249 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 167 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 28 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 52 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 92 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 6 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 11 T + 91 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 119 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 9 T + 47 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7 T + 141 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T - 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 184 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−15.7636623962, −15.2911860564, −14.9006276706, −13.9495920944, −13.1101943982, −12.7903791968, −12.4225597834, −12.2129487384, −11.4144335250, −11.3111199844, −10.7599170963, −10.4510434288, −9.76942346115, −9.51482842455, −8.84264745186, −8.23640530905, −7.71774681201, −7.08445691841, −6.61326826555, −6.56073990121, −5.49386450598, −4.82295754428, −4.39975574061, −3.64917370799, −2.61582879359, 0, 0, 0,
2.61582879359, 3.64917370799, 4.39975574061, 4.82295754428, 5.49386450598, 6.56073990121, 6.61326826555, 7.08445691841, 7.71774681201, 8.23640530905, 8.84264745186, 9.51482842455, 9.76942346115, 10.4510434288, 10.7599170963, 11.3111199844, 11.4144335250, 12.2129487384, 12.4225597834, 12.7903791968, 13.1101943982, 13.9495920944, 14.9006276706, 15.2911860564, 15.7636623962
