| L(s) = 1 | − 3-s − 4-s − 3·5-s − 4·7-s + 2·8-s − 2·9-s − 11-s + 12-s + 3·15-s + 16-s − 4·17-s + 3·20-s + 4·21-s − 5·23-s − 2·24-s + 8·25-s + 5·27-s + 4·28-s + 2·29-s − 4·31-s − 4·32-s + 33-s + 12·35-s + 2·36-s + 9·37-s − 6·40-s − 14·41-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 1.34·5-s − 1.51·7-s + 0.707·8-s − 2/3·9-s − 0.301·11-s + 0.288·12-s + 0.774·15-s + 1/4·16-s − 0.970·17-s + 0.670·20-s + 0.872·21-s − 1.04·23-s − 0.408·24-s + 8/5·25-s + 0.962·27-s + 0.755·28-s + 0.371·29-s − 0.718·31-s − 0.707·32-s + 0.174·33-s + 2.02·35-s + 1/3·36-s + 1.47·37-s − 0.948·40-s − 2.18·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6210 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6210 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| good | 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 105 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 45 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 61 | $C_4$ | \( 1 + T + 41 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 35 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 2 T - 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 73 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11 T + 63 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−17.5096038442, −16.6829987354, −16.4256186406, −16.1147428054, −15.6515733060, −14.9410248828, −14.5413499201, −13.7432428270, −13.2226485607, −12.8751175326, −12.2666588582, −11.6619775288, −11.3205074069, −10.4803509688, −10.2336546711, −9.35847386085, −8.71509548997, −8.21069803767, −7.45255192057, −6.80765574242, −6.17867418193, −5.31643943327, −4.45294608245, −3.80151105359, −2.86771368359, 0,
2.86771368359, 3.80151105359, 4.45294608245, 5.31643943327, 6.17867418193, 6.80765574242, 7.45255192057, 8.21069803767, 8.71509548997, 9.35847386085, 10.2336546711, 10.4803509688, 11.3205074069, 11.6619775288, 12.2666588582, 12.8751175326, 13.2226485607, 13.7432428270, 14.5413499201, 14.9410248828, 15.6515733060, 16.1147428054, 16.4256186406, 16.6829987354, 17.5096038442
