| L(s) = 1 | − 2-s − 4-s − 3·7-s + 3·8-s − 4·9-s + 3·14-s − 16-s + 9·17-s + 4·18-s − 8·25-s + 3·28-s + 11·31-s − 5·32-s − 9·34-s + 4·36-s − 4·41-s + 2·47-s + 6·49-s + 8·50-s − 9·56-s − 11·62-s + 12·63-s + 7·64-s − 9·68-s + 17·71-s − 12·72-s + 5·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.13·7-s + 1.06·8-s − 4/3·9-s + 0.801·14-s − 1/4·16-s + 2.18·17-s + 0.942·18-s − 8/5·25-s + 0.566·28-s + 1.97·31-s − 0.883·32-s − 1.54·34-s + 2/3·36-s − 0.624·41-s + 0.291·47-s + 6/7·49-s + 1.13·50-s − 1.20·56-s − 1.39·62-s + 1.51·63-s + 7/8·64-s − 1.09·68-s + 2.01·71-s − 1.41·72-s + 0.585·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5652819651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5652819651\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 57 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−9.977724437567373487557901492673, −9.832437681113807693736097851621, −9.379778314949305570547510066190, −8.714582214203457968332297297812, −8.192506058363363009683592055373, −7.88191958163304447946115310907, −7.31060466326429919864614643728, −6.39309172282255549188864997571, −6.00175177942932547627619954337, −5.38762159975313116979896564332, −4.78066567311042312780984582619, −3.62336072995021998096037357227, −3.40439014602821561225859213935, −2.33470701611076728672818862733, −0.76107059601051481081589245621,
0.76107059601051481081589245621, 2.33470701611076728672818862733, 3.40439014602821561225859213935, 3.62336072995021998096037357227, 4.78066567311042312780984582619, 5.38762159975313116979896564332, 6.00175177942932547627619954337, 6.39309172282255549188864997571, 7.31060466326429919864614643728, 7.88191958163304447946115310907, 8.192506058363363009683592055373, 8.714582214203457968332297297812, 9.379778314949305570547510066190, 9.832437681113807693736097851621, 9.977724437567373487557901492673
