L(s) = 1 | − 4-s + 2·9-s + 6·11-s + 16-s − 2·19-s + 12·29-s + 8·31-s − 2·36-s − 6·41-s − 6·44-s + 8·61-s − 64-s + 24·71-s + 2·76-s + 20·79-s − 5·81-s + 12·89-s + 12·99-s + 24·101-s + 8·109-s − 12·116-s + 5·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 2/3·9-s + 1.80·11-s + 1/4·16-s − 0.458·19-s + 2.22·29-s + 1.43·31-s − 1/3·36-s − 0.937·41-s − 0.904·44-s + 1.02·61-s − 1/8·64-s + 2.84·71-s + 0.229·76-s + 2.25·79-s − 5/9·81-s + 1.27·89-s + 1.20·99-s + 2.38·101-s + 0.766·109-s − 1.11·116-s + 5/11·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.306216117\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.306216117\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.138744981811695076919354342224, −8.652910749363926131509156124292, −8.523311681177393412106236611631, −8.033323737631820686516555539887, −7.71751083306575951341362370884, −7.12410372081235850158511294763, −6.56172867192083274096652704050, −6.52272979868623536037539665570, −6.34445543040135215600134888180, −5.61595208159106136760849777942, −4.89344374758723570901555445984, −4.87431158055118309447736737843, −4.37362738480488404715608122269, −3.80040955660923719216990012584, −3.66923422016633948060656835269, −3.01706203168949667323974322938, −2.35016083738854061286538038242, −1.84067163659011269625107824735, −1.02235225031887039549613264526, −0.798202320613114203332365980096,
0.798202320613114203332365980096, 1.02235225031887039549613264526, 1.84067163659011269625107824735, 2.35016083738854061286538038242, 3.01706203168949667323974322938, 3.66923422016633948060656835269, 3.80040955660923719216990012584, 4.37362738480488404715608122269, 4.87431158055118309447736737843, 4.89344374758723570901555445984, 5.61595208159106136760849777942, 6.34445543040135215600134888180, 6.52272979868623536037539665570, 6.56172867192083274096652704050, 7.12410372081235850158511294763, 7.71751083306575951341362370884, 8.033323737631820686516555539887, 8.523311681177393412106236611631, 8.652910749363926131509156124292, 9.138744981811695076919354342224