| L(s) = 1 | + 1.41·2-s − 1.73i·3-s + 2.00·4-s − 2.44i·6-s + (−1.04 + 6.92i)7-s + 2.82·8-s − 2.99·9-s − 15.9·11-s − 3.46i·12-s − 6.19i·13-s + (−1.47 + 9.78i)14-s + 4.00·16-s + 19.7i·17-s − 4.24·18-s − 29.3i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.500·4-s − 0.408i·6-s + (−0.149 + 0.988i)7-s + 0.353·8-s − 0.333·9-s − 1.45·11-s − 0.288i·12-s − 0.476i·13-s + (−0.105 + 0.699i)14-s + 0.250·16-s + 1.16i·17-s − 0.235·18-s − 1.54i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1580651423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1580651423\) |
| \(L(2)\) |
|
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 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.04 - 6.92i)T \) |
| good | 11 | \( 1 + 15.9T + 121T^{2} \) |
| 13 | \( 1 + 6.19iT - 169T^{2} \) |
| 17 | \( 1 - 19.7iT - 289T^{2} \) |
| 19 | \( 1 + 29.3iT - 361T^{2} \) |
| 23 | \( 1 + 18.3T + 529T^{2} \) |
| 29 | \( 1 + 53.2T + 841T^{2} \) |
| 31 | \( 1 + 37.4iT - 961T^{2} \) |
| 37 | \( 1 + 61.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 21.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 25.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 10.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 64.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 18.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 91.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 18.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 108.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 73.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 57.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 36.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 133. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 29.2iT - 9.40e3T^{2} \) |
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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
−9.181613258276343031894446613641, −8.278745584222699433666261229945, −7.59187689694399918407131555610, −6.63012473024252690223489576419, −5.60013458448656299983647644905, −5.30536981481385806862710982873, −3.85654314532204115835280350785, −2.69171018090902507553939322730, −2.01778182107775367159983756084, −0.03350520790119909361015398347,
1.89097411485181143356068376604, 3.24612109660514772132828321125, 3.92648011048276466294568245920, 4.99871458965605549169389025153, 5.57666020332097421404922609604, 6.79544658719943008244986588608, 7.55340912117476244126492234994, 8.347704199570498903832819414949, 9.646994037085549084916695958418, 10.23245728510742204568489640295
