L(s) = 1 | + (0.948 − 1.04i)2-s − 1.76i·3-s + (−0.201 − 1.98i)4-s + (−1.85 + 1.24i)5-s + (−1.84 − 1.67i)6-s − 4.29·7-s + (−2.27 − 1.67i)8-s − 0.102·9-s + (−0.457 + 3.12i)10-s + 0.376i·11-s + (−3.50 + 0.355i)12-s − 0.928·13-s + (−4.07 + 4.50i)14-s + (2.19 + 3.27i)15-s + (−3.91 + 0.802i)16-s − 5.96i·17-s + ⋯ |
L(s) = 1 | + (0.670 − 0.741i)2-s − 1.01i·3-s + (−0.100 − 0.994i)4-s + (−0.831 + 0.556i)5-s + (−0.754 − 0.681i)6-s − 1.62·7-s + (−0.805 − 0.592i)8-s − 0.0343·9-s + (−0.144 + 0.989i)10-s + 0.113i·11-s + (−1.01 + 0.102i)12-s − 0.257·13-s + (−1.08 + 1.20i)14-s + (0.565 + 0.845i)15-s + (−0.979 + 0.200i)16-s − 1.44i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.146122 + 1.01144i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.146122 + 1.01144i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.948 + 1.04i)T \) |
| 5 | \( 1 + (1.85 - 1.24i)T \) |
| 19 | \( 1 + (-3.61 + 2.43i)T \) |
good | 3 | \( 1 + 1.76iT - 3T^{2} \) |
| 7 | \( 1 + 4.29T + 7T^{2} \) |
| 11 | \( 1 - 0.376iT - 11T^{2} \) |
| 13 | \( 1 + 0.928T + 13T^{2} \) |
| 17 | \( 1 + 5.96iT - 17T^{2} \) |
| 23 | \( 1 - 0.352T + 23T^{2} \) |
| 29 | \( 1 + 3.89iT - 29T^{2} \) |
| 31 | \( 1 + 1.06T + 31T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 - 6.82iT - 41T^{2} \) |
| 43 | \( 1 + 9.45T + 43T^{2} \) |
| 47 | \( 1 + 2.56T + 47T^{2} \) |
| 53 | \( 1 + 3.18T + 53T^{2} \) |
| 59 | \( 1 + 3.98T + 59T^{2} \) |
| 61 | \( 1 + 3.22T + 61T^{2} \) |
| 67 | \( 1 + 8.75iT - 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 5.09iT - 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 3.49T + 83T^{2} \) |
| 89 | \( 1 + 10.4iT - 89T^{2} \) |
| 97 | \( 1 - 9.74T + 97T^{2} \) |
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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
−11.24467674946360883210480190981, −9.925469400509395868232084531655, −9.439075868379359570657638485261, −7.72028529631021871205413852317, −6.85021997066199863431512595967, −6.28080389026347401345145690926, −4.72190960793149003677033556448, −3.36361097609758642488905467065, −2.60338768798564933684100978192, −0.54134650920911965996824263592,
3.36547738087746694480883479627, 3.82445561287559462925717140325, 4.92482288842535926612269952373, 5.99549314831292778009417180221, 7.03650632435563097360861847654, 8.091075507142264946318238088381, 9.110171759707514991220889914967, 9.838183944727375832160457726381, 10.94736678381403410601072643334, 12.20258932924902816894236352521