| L(s) = 1 | − 3-s − 4-s − 5-s − 3·7-s + 2·8-s − 2·9-s + 5·11-s + 12-s − 3·13-s + 15-s + 16-s − 2·17-s + 20-s + 3·21-s + 8·23-s − 2·24-s + 25-s + 2·27-s + 3·28-s + 29-s − 31-s − 4·32-s − 5·33-s + 3·35-s + 2·36-s − 4·37-s + 3·39-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 0.447·5-s − 1.13·7-s + 0.707·8-s − 2/3·9-s + 1.50·11-s + 0.288·12-s − 0.832·13-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.223·20-s + 0.654·21-s + 1.66·23-s − 0.408·24-s + 1/5·25-s + 0.384·27-s + 0.566·28-s + 0.185·29-s − 0.179·31-s − 0.707·32-s − 0.870·33-s + 0.507·35-s + 1/3·36-s − 0.657·37-s + 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 834 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 834 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3676098807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3676098807\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 139 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 12 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 15 T + 186 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 9 T + 30 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 7 T + 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 112 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−19.5498310312, −19.3435842630, −18.8747556104, −17.9344929515, −17.1949405684, −17.0187543082, −16.5276879710, −15.8754941792, −14.8988880324, −14.6818567971, −13.6718238204, −13.3213654465, −12.3808780905, −12.0170264206, −11.2183848234, −10.6046936283, −9.66599640236, −9.11838420690, −8.39429356341, −7.08082152627, −6.71343196046, −5.52246161490, −4.53472704076, −3.38079152731,
3.38079152731, 4.53472704076, 5.52246161490, 6.71343196046, 7.08082152627, 8.39429356341, 9.11838420690, 9.66599640236, 10.6046936283, 11.2183848234, 12.0170264206, 12.3808780905, 13.3213654465, 13.6718238204, 14.6818567971, 14.8988880324, 15.8754941792, 16.5276879710, 17.0187543082, 17.1949405684, 17.9344929515, 18.8747556104, 19.3435842630, 19.5498310312
