L(s) = 1 | + (1.71 − 0.213i)3-s + (−2.07 + 3.58i)5-s + (2.19 − 1.48i)7-s + (2.90 − 0.733i)9-s + (−0.434 − 0.752i)11-s + (2.86 + 4.95i)13-s + (−2.79 + 6.61i)15-s + (−1.44 + 2.50i)17-s + (−2.00 − 3.47i)19-s + (3.44 − 3.01i)21-s + (2.91 − 5.04i)23-s + (−6.09 − 10.5i)25-s + (4.84 − 1.88i)27-s + (−0.900 + 1.55i)29-s − 2.96·31-s + ⋯ |
L(s) = 1 | + (0.992 − 0.123i)3-s + (−0.926 + 1.60i)5-s + (0.827 − 0.561i)7-s + (0.969 − 0.244i)9-s + (−0.130 − 0.226i)11-s + (0.793 + 1.37i)13-s + (−0.722 + 1.70i)15-s + (−0.350 + 0.607i)17-s + (−0.460 − 0.797i)19-s + (0.752 − 0.658i)21-s + (0.607 − 1.05i)23-s + (−1.21 − 2.11i)25-s + (0.932 − 0.362i)27-s + (−0.167 + 0.289i)29-s − 0.531·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.53040 + 0.477333i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53040 + 0.477333i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.71 + 0.213i)T \) |
| 7 | \( 1 + (-2.19 + 1.48i)T \) |
good | 5 | \( 1 + (2.07 - 3.58i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.434 + 0.752i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.86 - 4.95i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.44 - 2.50i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.00 + 3.47i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.91 + 5.04i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.900 - 1.55i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 + (2.64 + 4.58i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.89 + 10.2i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.00 - 3.47i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 2.34T + 47T^{2} \) |
| 53 | \( 1 + (1.09 - 1.88i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 3.05T + 59T^{2} \) |
| 61 | \( 1 - 5.63T + 61T^{2} \) |
| 67 | \( 1 + 2.51T + 67T^{2} \) |
| 71 | \( 1 + 1.09T + 71T^{2} \) |
| 73 | \( 1 + (-0.723 + 1.25i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 2.12T + 79T^{2} \) |
| 83 | \( 1 + (-2.18 + 3.78i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.83 - 10.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.98 - 6.90i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.96966158004322580220368434303, −10.87528680138129210113636719149, −10.66062985001933157335379874742, −8.989636457879397388007437785759, −8.196408857460836909106828573140, −7.12721263774478013630209403178, −6.66548044285561418678378993543, −4.33278895680650272538757719195, −3.57092669392556173013210634043, −2.17092269136658864600564506600,
1.47866794895013501675257818653, 3.39761808121201185183701185234, 4.59193466641269487303258694019, 5.43961437672076612650852033176, 7.53455247616693673178709490859, 8.280085471352558589152503491741, 8.694956187875913890297038527000, 9.765863227627013955358917185890, 11.13877446791637849264952885197, 12.10335299438877830656554477737