L(s) = 1 | − 1.03·5-s + (1.07 + 2.41i)7-s + 1.58·11-s + (−2.52 − 4.37i)13-s + (2.58 + 4.47i)17-s + (−0.392 + 0.680i)19-s + 5.86·23-s − 3.93·25-s + (−4.44 + 7.69i)29-s + (0.575 − 0.996i)31-s + (−1.10 − 2.49i)35-s + (4.07 − 7.06i)37-s + (3.87 + 6.70i)41-s + (1.26 − 2.19i)43-s + (4.24 + 7.35i)47-s + ⋯ |
L(s) = 1 | − 0.461·5-s + (0.406 + 0.913i)7-s + 0.478·11-s + (−0.700 − 1.21i)13-s + (0.626 + 1.08i)17-s + (−0.0901 + 0.156i)19-s + 1.22·23-s − 0.787·25-s + (−0.825 + 1.42i)29-s + (0.103 − 0.179i)31-s + (−0.187 − 0.421i)35-s + (0.670 − 1.16i)37-s + (0.604 + 1.04i)41-s + (0.193 − 0.334i)43-s + (0.619 + 1.07i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.462300664\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.462300664\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.07 - 2.41i)T \) |
good | 5 | \( 1 + 1.03T + 5T^{2} \) |
| 11 | \( 1 - 1.58T + 11T^{2} \) |
| 13 | \( 1 + (2.52 + 4.37i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.58 - 4.47i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.392 - 0.680i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 5.86T + 23T^{2} \) |
| 29 | \( 1 + (4.44 - 7.69i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.575 + 0.996i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.07 + 7.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.87 - 6.70i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.26 + 2.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.24 - 7.35i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.41 + 4.17i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.93 + 3.35i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.82 - 8.35i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.837 + 1.45i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + (-3.04 - 5.27i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.15 - 7.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.12 - 12.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.69 - 11.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.67 - 4.63i)T + (-48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.170689268268991289645887573053, −8.304692772459750953757529965607, −7.79275337370423446079165456318, −6.98641815997848514179628180834, −5.79171285864513226754401302781, −5.43710614331438006660001714598, −4.34259476407475790412825133397, −3.39864101284060794111228722929, −2.47886477512310331679170631433, −1.21171569968898655456573410304,
0.56004983034709982358861311727, 1.85064183248297682996548616088, 3.08661869101925171842697342030, 4.18264119980548576181467316797, 4.58674127748438483561225788122, 5.66344151832471422421904896651, 6.79666120724157603933538752430, 7.31225487005508997764814627593, 7.88779873355972529698243842802, 8.963158973747563615908361811548