L(s) = 1 | + (−1.39 + 0.221i)2-s + (−2.48 + 1.80i)3-s + (1.90 − 0.618i)4-s − 1.61·5-s + (3.07 − 3.07i)6-s + (0.224 − 0.0729i)7-s + (−2.52 + 1.28i)8-s + (1.99 − 6.15i)9-s + (2.26 − 0.357i)10-s + (−1.90 − 5.85i)11-s + (−3.61 + 4.97i)12-s + (−0.263 − 0.363i)13-s + (−0.297 + 0.151i)14-s + (4.02 − 2.92i)15-s + (3.23 − 2.35i)16-s + (−1.54 − 0.502i)17-s + ⋯ |
L(s) = 1 | + (−0.987 + 0.156i)2-s + (−1.43 + 1.04i)3-s + (0.951 − 0.309i)4-s − 0.723·5-s + (1.25 − 1.25i)6-s + (0.0848 − 0.0275i)7-s + (−0.891 + 0.453i)8-s + (0.666 − 2.05i)9-s + (0.714 − 0.113i)10-s + (−0.573 − 1.76i)11-s + (−1.04 + 1.43i)12-s + (−0.0732 − 0.100i)13-s + (−0.0795 + 0.0405i)14-s + (1.04 − 0.755i)15-s + (0.809 − 0.587i)16-s + (−0.374 − 0.121i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0563782 - 0.0788195i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0563782 - 0.0788195i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.39 - 0.221i)T \) |
| 31 | \( 1 + (1.76 + 5.28i)T \) |
good | 3 | \( 1 + (2.48 - 1.80i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 + (-0.224 + 0.0729i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (1.90 + 5.85i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.263 + 0.363i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.54 + 0.502i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (2.93 - 4.04i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.224 + 0.690i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (4.30 - 5.93i)T + (-8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + 9.23iT - 37T^{2} \) |
| 41 | \( 1 + (-4.23 - 3.07i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (2.12 + 1.54i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (3.30 + 4.54i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (8.78 + 2.85i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.21 - 4.42i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + 0.898iT - 61T^{2} \) |
| 67 | \( 1 - 5.76iT - 67T^{2} \) |
| 71 | \( 1 + (-6.29 - 2.04i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.80 - 0.587i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.07 + 6.38i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-5.34 - 3.88i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (2.33 - 0.759i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.454 - 1.40i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−12.64028469514367913192375147902, −11.33765907431526804935401746670, −11.13554602148151672786999941741, −10.20010310353925893445491262629, −8.983846161139569866985002760426, −7.81676856566750777767254676717, −6.26383564222240616521385959331, −5.43440821100125345400309393070, −3.70293624395233719707274430253, −0.15855955388960393802447133357,
1.96446641156117055601860268381, 4.78447691672486097476754939985, 6.42580932404463605550462538552, 7.25412453127035627731006535825, 8.016219053947058930015526912100, 9.737265820176241624672846405988, 10.87832866423630376061455977114, 11.58287834383934598695012932900, 12.40825562727354198420379350493, 13.07259498406830053352695974432