L(s) = 1 | + 1.41i·2-s + (2.46 − 1.70i)3-s − 2.00·4-s + (2.41 + 3.48i)6-s + 2.64·7-s − 2.82i·8-s + (3.15 − 8.42i)9-s − 2.66i·11-s + (−4.93 + 3.41i)12-s − 5.56·13-s + 3.74i·14-s + 4.00·16-s − 18.5i·17-s + (11.9 + 4.46i)18-s − 0.634·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.821 − 0.569i)3-s − 0.500·4-s + (0.402 + 0.581i)6-s + 0.377·7-s − 0.353i·8-s + (0.350 − 0.936i)9-s − 0.242i·11-s + (−0.410 + 0.284i)12-s − 0.427·13-s + 0.267i·14-s + 0.250·16-s − 1.09i·17-s + (0.662 + 0.248i)18-s − 0.0334·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.172656004\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.172656004\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (-2.46 + 1.70i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
good | 11 | \( 1 + 2.66iT - 121T^{2} \) |
| 13 | \( 1 + 5.56T + 169T^{2} \) |
| 17 | \( 1 + 18.5iT - 289T^{2} \) |
| 19 | \( 1 + 0.634T + 361T^{2} \) |
| 23 | \( 1 - 15.6iT - 529T^{2} \) |
| 29 | \( 1 + 43.3iT - 841T^{2} \) |
| 31 | \( 1 - 13.5T + 961T^{2} \) |
| 37 | \( 1 - 35.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 15.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 64.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 52.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 51.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 32.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 104.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 113.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 36.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 144.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 57.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 45.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 80.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 96.2T + 9.40e3T^{2} \) |
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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
−9.477349599239005501054982876648, −8.525432183017247443367565745015, −7.904427062357952587163648496735, −7.20464368695726807743165450333, −6.41928917226572444525215166894, −5.36856664617443002031608141597, −4.35536018461000049620517044729, −3.24200934531030027214328859158, −2.10056124566850806305685245734, −0.61312679515156746603601382570,
1.47797811211788380873984809914, 2.51967297366802844004819358630, 3.49352084998109541030525644570, 4.45229248893827493109825928485, 5.11934503358583359677500155171, 6.48196656851558372048383520018, 7.72689234924923464272803224135, 8.333629833592557722512144858236, 9.140489961621777316996736724922, 9.870645084446965077485460058956