| L(s) = 1 | − 2·4-s − 8·9-s − 2·25-s + 16·31-s + 16·36-s + 16·49-s + 8·64-s − 48·71-s − 16·79-s + 30·81-s + 24·89-s + 4·100-s + 20·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4-s − 8/3·9-s − 2/5·25-s + 2.87·31-s + 8/3·36-s + 16/7·49-s + 64-s − 5.69·71-s − 1.80·79-s + 10/3·81-s + 2.54·89-s + 2/5·100-s + 1.81·121-s − 2.87·124-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2834354529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2834354529\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−12.18709499912001420814971733755, −11.96711470515038797106701802629, −11.45864479324728532217164289160, −11.35752496417492728107494260346, −11.21121993960389503000334126595, −10.35701049142237880241621067005, −10.26052463905149426252027589742, −10.11235888394124421199352608016, −9.528438312547950137029041324522, −8.934312913185403525461999868856, −8.798554870288155207523017078000, −8.688515774099160318825793739805, −8.461263238288824085470361171638, −7.70014586697367922220762333574, −7.66792177713269141433414264714, −7.05017351554440068656211246655, −6.24336100117680943233153123675, −6.20129881988782172981086432450, −5.70799407082769302029455829041, −5.38378164260750889619052821507, −4.67560823073527008864014994875, −4.46317149075080133824051236363, −3.73298347191104287828049766918, −2.85942685608777736062149397746, −2.68614116154331665113960259570,
2.68614116154331665113960259570, 2.85942685608777736062149397746, 3.73298347191104287828049766918, 4.46317149075080133824051236363, 4.67560823073527008864014994875, 5.38378164260750889619052821507, 5.70799407082769302029455829041, 6.20129881988782172981086432450, 6.24336100117680943233153123675, 7.05017351554440068656211246655, 7.66792177713269141433414264714, 7.70014586697367922220762333574, 8.461263238288824085470361171638, 8.688515774099160318825793739805, 8.798554870288155207523017078000, 8.934312913185403525461999868856, 9.528438312547950137029041324522, 10.11235888394124421199352608016, 10.26052463905149426252027589742, 10.35701049142237880241621067005, 11.21121993960389503000334126595, 11.35752496417492728107494260346, 11.45864479324728532217164289160, 11.96711470515038797106701802629, 12.18709499912001420814971733755
