| L(s) = 1 | + 2-s − 2·3-s − 4-s − 2·6-s − 4·7-s − 8-s + 2·9-s + 3·11-s + 2·12-s − 3·13-s − 4·14-s − 16-s − 2·17-s + 2·18-s − 4·19-s + 8·21-s + 3·22-s − 9·23-s + 2·24-s − 2·25-s − 3·26-s − 6·27-s + 4·28-s + 7·29-s − 3·31-s − 5·32-s − 6·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.816·6-s − 1.51·7-s − 0.353·8-s + 2/3·9-s + 0.904·11-s + 0.577·12-s − 0.832·13-s − 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.471·18-s − 0.917·19-s + 1.74·21-s + 0.639·22-s − 1.87·23-s + 0.408·24-s − 2/5·25-s − 0.588·26-s − 1.15·27-s + 0.755·28-s + 1.29·29-s − 0.538·31-s − 0.883·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14147 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14147 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T - 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 21 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 48 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 10 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T - 107 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 97 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 5 T + 10 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 13 T + 125 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−16.5086663155, −16.0228058645, −15.4355710725, −15.0138595146, −14.2610942966, −13.9513906657, −13.4282078694, −12.8165932822, −12.6044352076, −12.1010670024, −11.5695859215, −11.0989446802, −10.2361153158, −9.98076500817, −9.35583044229, −8.94416024707, −8.00899919303, −7.25468696647, −6.51206047219, −6.18440962837, −5.68817541851, −4.75187486458, −4.19204694334, −3.68709454830, −2.32366164333, 0,
2.32366164333, 3.68709454830, 4.19204694334, 4.75187486458, 5.68817541851, 6.18440962837, 6.51206047219, 7.25468696647, 8.00899919303, 8.94416024707, 9.35583044229, 9.98076500817, 10.2361153158, 11.0989446802, 11.5695859215, 12.1010670024, 12.6044352076, 12.8165932822, 13.4282078694, 13.9513906657, 14.2610942966, 15.0138595146, 15.4355710725, 16.0228058645, 16.5086663155
