| L(s) = 1 | − 4.11·3-s + 1.10·5-s + (3.73 + 5.92i)7-s + 7.92·9-s − 10.7i·11-s − 13.1·13-s − 4.53·15-s − 20.5i·17-s − 24.3·19-s + (−15.3 − 24.3i)21-s − 16.7·23-s − 23.7·25-s + 4.40·27-s − 4.21i·29-s − 52.8i·31-s + ⋯ |
| L(s) = 1 | − 1.37·3-s + 0.220·5-s + (0.533 + 0.846i)7-s + 0.880·9-s − 0.976i·11-s − 1.01·13-s − 0.302·15-s − 1.20i·17-s − 1.28·19-s + (−0.731 − 1.16i)21-s − 0.729·23-s − 0.951·25-s + 0.163·27-s − 0.145i·29-s − 1.70i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.120114 - 0.306781i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.120114 - 0.306781i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-3.73 - 5.92i)T \) |
| good | 3 | \( 1 + 4.11T + 9T^{2} \) |
| 5 | \( 1 - 1.10T + 25T^{2} \) |
| 11 | \( 1 + 10.7iT - 121T^{2} \) |
| 13 | \( 1 + 13.1T + 169T^{2} \) |
| 17 | \( 1 + 20.5iT - 289T^{2} \) |
| 19 | \( 1 + 24.3T + 361T^{2} \) |
| 23 | \( 1 + 16.7T + 529T^{2} \) |
| 29 | \( 1 + 4.21iT - 841T^{2} \) |
| 31 | \( 1 + 52.8iT - 961T^{2} \) |
| 37 | \( 1 - 9.97iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 23.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 65.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 52.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 47.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 40.2T + 3.48e3T^{2} \) |
| 61 | \( 1 - 1.10T + 3.72e3T^{2} \) |
| 67 | \( 1 - 65.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 113.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 91.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 27.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 2.34T + 6.88e3T^{2} \) |
| 89 | \( 1 + 50.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 74.2iT - 9.40e3T^{2} \) |
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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
−11.72030631261205226551515121087, −10.93949415493706982669240582128, −9.882654618953570933721899537418, −8.754875944451623298184185929084, −7.51990818828638719624303833142, −6.13001176387194351452408783314, −5.57559987010999020952643238347, −4.47554505877431418901836369131, −2.35885627458926591875928123026, −0.20456342672528689966014514594,
1.76409880698915295936708820887, 4.20241434391607617948226915762, 5.04724997615210161116652987192, 6.25764542563611978587808199498, 7.13602931285980426453247964442, 8.298092420058570172954243411705, 9.994754747722335779455257500114, 10.43509699995737761927989840658, 11.41587472241298652845361929784, 12.32181126467880361940907601173
