| L(s) = 1 | + 2-s + 4-s − 2·5-s + 3·8-s − 2·10-s − 8·11-s + 4·13-s + 16-s − 6·17-s − 2·19-s − 2·20-s − 8·22-s + 7·23-s + 3·25-s + 4·26-s − 7·29-s + 6·31-s − 32-s − 6·34-s + 20·37-s − 2·38-s − 6·40-s + 7·41-s + 4·43-s − 8·44-s + 7·46-s + 16·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 1.06·8-s − 0.632·10-s − 2.41·11-s + 1.10·13-s + 1/4·16-s − 1.45·17-s − 0.458·19-s − 0.447·20-s − 1.70·22-s + 1.45·23-s + 3/5·25-s + 0.784·26-s − 1.29·29-s + 1.07·31-s − 0.176·32-s − 1.02·34-s + 3.28·37-s − 0.324·38-s − 0.948·40-s + 1.09·41-s + 0.609·43-s − 1.20·44-s + 1.03·46-s + 2.33·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16040025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16040025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.238530310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.238530310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 89 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 56 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 9 T + 134 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 58 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 120 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 9 T + 174 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 130 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 21 T + 300 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
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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
−8.328936450801043936960407431008, −8.082543720997742921131271610236, −8.017446034407653351427053992497, −7.52109710238571312676548393743, −7.21854688754628763560709392105, −6.80092106244138217634256140780, −6.49068438221811120342017618777, −5.94695147108400955989122521997, −5.51152717763165398755600340080, −5.38763995474000721144921252503, −4.62319421956852696503660116069, −4.51890888248058140961462608768, −4.02581030922650304195741025443, −4.01866447218185168617495396770, −2.92151117179989458900730836725, −2.69776460379532864780015231951, −2.65058487288545241990042680997, −1.82816874827992887201473975198, −1.11936254025231139229459391314, −0.40681170880790860119156690916,
0.40681170880790860119156690916, 1.11936254025231139229459391314, 1.82816874827992887201473975198, 2.65058487288545241990042680997, 2.69776460379532864780015231951, 2.92151117179989458900730836725, 4.01866447218185168617495396770, 4.02581030922650304195741025443, 4.51890888248058140961462608768, 4.62319421956852696503660116069, 5.38763995474000721144921252503, 5.51152717763165398755600340080, 5.94695147108400955989122521997, 6.49068438221811120342017618777, 6.80092106244138217634256140780, 7.21854688754628763560709392105, 7.52109710238571312676548393743, 8.017446034407653351427053992497, 8.082543720997742921131271610236, 8.328936450801043936960407431008
