| L(s) = 1 | − 4·2-s + 2·4-s + 20·8-s − 45·16-s − 16·17-s + 10·25-s − 16·32-s + 64·34-s + 16·47-s + 16·49-s − 40·50-s + 204·64-s − 32·68-s − 64·94-s − 64·98-s + 20·100-s + 32·109-s − 8·113-s + 32·121-s + 127-s − 232·128-s + 131-s − 320·136-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4-s + 7.07·8-s − 11.2·16-s − 3.88·17-s + 2·25-s − 2.82·32-s + 10.9·34-s + 2.33·47-s + 16/7·49-s − 5.65·50-s + 51/2·64-s − 3.88·68-s − 6.60·94-s − 6.46·98-s + 2·100-s + 3.06·109-s − 0.752·113-s + 2.90·121-s + 0.0887·127-s − 20.5·128-s + 0.0873·131-s − 27.4·136-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2644952650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2644952650\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.15535387817615218177053867577, −6.89441198002620944479204681991, −6.64819501240829498313412188281, −6.58766553545116715499515231749, −5.93994157065901693835617719328, −5.61844108902871049430374794128, −5.58651420708566271968837497144, −5.52471299563683930282449120043, −4.80633727248168725455122522462, −4.67453557899356843902531008749, −4.54692690524426053449053750168, −4.51375550186986941440144821891, −4.45878672691699677373787573784, −3.98944773247433625283473496492, −3.70744149618744089772675527515, −3.56969072108167772219068881157, −3.06473812396423715640600791498, −2.61938015957899748779785206780, −2.18471526319002699630922473079, −2.06008500243619462560099847675, −1.84263635487492244373879032602, −1.27850485289891204483524227518, −0.862428113003867811530034793512, −0.57269753480208290192846670141, −0.41592243861125910798537302754,
0.41592243861125910798537302754, 0.57269753480208290192846670141, 0.862428113003867811530034793512, 1.27850485289891204483524227518, 1.84263635487492244373879032602, 2.06008500243619462560099847675, 2.18471526319002699630922473079, 2.61938015957899748779785206780, 3.06473812396423715640600791498, 3.56969072108167772219068881157, 3.70744149618744089772675527515, 3.98944773247433625283473496492, 4.45878672691699677373787573784, 4.51375550186986941440144821891, 4.54692690524426053449053750168, 4.67453557899356843902531008749, 4.80633727248168725455122522462, 5.52471299563683930282449120043, 5.58651420708566271968837497144, 5.61844108902871049430374794128, 5.93994157065901693835617719328, 6.58766553545116715499515231749, 6.64819501240829498313412188281, 6.89441198002620944479204681991, 7.15535387817615218177053867577
