| L(s) = 1 | + 1.41·2-s + 2.00·4-s + 2.23i·5-s + 8.24i·7-s + 2.82·8-s + 3.16i·10-s − 15.8i·11-s − 14.3·13-s + 11.6i·14-s + 4.00·16-s + 10.1i·17-s − 36.5i·19-s + 4.47i·20-s − 22.3i·22-s + (−22.2 − 5.84i)23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + 0.447i·5-s + 1.17i·7-s + 0.353·8-s + 0.316i·10-s − 1.43i·11-s − 1.10·13-s + 0.832i·14-s + 0.250·16-s + 0.598i·17-s − 1.92i·19-s + 0.223i·20-s − 1.01i·22-s + (−0.967 − 0.254i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.458279011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.458279011\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (22.2 + 5.84i)T \) |
| good | 7 | \( 1 - 8.24iT - 49T^{2} \) |
| 11 | \( 1 + 15.8iT - 121T^{2} \) |
| 13 | \( 1 + 14.3T + 169T^{2} \) |
| 17 | \( 1 - 10.1iT - 289T^{2} \) |
| 19 | \( 1 + 36.5iT - 361T^{2} \) |
| 29 | \( 1 + 6.46T + 841T^{2} \) |
| 31 | \( 1 + 42.8T + 961T^{2} \) |
| 37 | \( 1 + 63.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 37.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 6.00iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 32.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 36.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 6.65T + 3.48e3T^{2} \) |
| 61 | \( 1 + 55.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.45iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 118.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 82.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 133. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 67.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 104. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 98.6iT - 9.40e3T^{2} \) |
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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
−8.886787944751236073889932169213, −7.78523174774798047368302719846, −7.08149879219165144456211528082, −6.01779246360324586730283051115, −5.69675496771400215261693988076, −4.73419649092426097346827313563, −3.67494304302941982701694730035, −2.72867960156740387078557549849, −2.15362041029897165381455950221, −0.25999374258309131221078614006,
1.37544054114727922248473650292, 2.28626442746301809573209323911, 3.67580434962290216254628715553, 4.27930521108234355640073730966, 5.00726856659527612295181644282, 5.85538734340873850007087383113, 6.97067325869759262311797417498, 7.47070971856622865607287322641, 8.049247549621604262463043792691, 9.435235387793497128197251119031
