L(s) = 1 | + 4·7-s − 5·9-s + 12·17-s − 6·23-s − 25-s + 10·31-s − 2·49-s − 20·63-s + 30·71-s − 8·73-s + 4·79-s + 16·81-s − 18·89-s − 14·97-s + 16·103-s − 30·113-s + 48·119-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 60·153-s + 157-s − 24·161-s + ⋯ |
L(s) = 1 | + 1.51·7-s − 5/3·9-s + 2.91·17-s − 1.25·23-s − 1/5·25-s + 1.79·31-s − 2/7·49-s − 2.51·63-s + 3.56·71-s − 0.936·73-s + 0.450·79-s + 16/9·81-s − 1.90·89-s − 1.42·97-s + 1.57·103-s − 2.82·113-s + 4.40·119-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 4.85·153-s + 0.0798·157-s − 1.89·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.373460837\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.373460837\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−10.54980513640919233558709119533, −9.813753565180213813561671136388, −9.700092895780422192241801282926, −8.674866558231347995249063814171, −8.132665807481574488057539990847, −8.050490063857036795242146503456, −7.60852684015633966980596452192, −6.51356460177744683595774086764, −5.96029809065759357817228429363, −5.29396314589658220886171477329, −5.11227132195889762323685903220, −4.04654564288201609443200871032, −3.27381985666804460223319703873, −2.49141172081529131249029798363, −1.28296422834420783986833535835,
1.28296422834420783986833535835, 2.49141172081529131249029798363, 3.27381985666804460223319703873, 4.04654564288201609443200871032, 5.11227132195889762323685903220, 5.29396314589658220886171477329, 5.96029809065759357817228429363, 6.51356460177744683595774086764, 7.60852684015633966980596452192, 8.050490063857036795242146503456, 8.132665807481574488057539990847, 8.674866558231347995249063814171, 9.700092895780422192241801282926, 9.813753565180213813561671136388, 10.54980513640919233558709119533