| L(s) = 1 | + 1.41i·2-s − 1.35·3-s − 2.00·4-s + 2.23·5-s − 1.92i·6-s + 12.4i·7-s − 2.82i·8-s − 7.15·9-s + 3.16i·10-s + (−7.23 + 8.28i)11-s + 2.71·12-s + 1.28i·13-s − 17.5·14-s − 3.03·15-s + 4.00·16-s − 3.33i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.452·3-s − 0.500·4-s + 0.447·5-s − 0.320i·6-s + 1.77i·7-s − 0.353i·8-s − 0.795·9-s + 0.316i·10-s + (−0.658 + 0.752i)11-s + 0.226·12-s + 0.0984i·13-s − 1.25·14-s − 0.202·15-s + 0.250·16-s − 0.195i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.332427 + 0.885477i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.332427 + 0.885477i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 5 | \( 1 - 2.23T \) |
| 11 | \( 1 + (7.23 - 8.28i)T \) |
| good | 3 | \( 1 + 1.35T + 9T^{2} \) |
| 7 | \( 1 - 12.4iT - 49T^{2} \) |
| 13 | \( 1 - 1.28iT - 169T^{2} \) |
| 17 | \( 1 + 3.33iT - 289T^{2} \) |
| 19 | \( 1 - 1.88iT - 361T^{2} \) |
| 23 | \( 1 - 32.3T + 529T^{2} \) |
| 29 | \( 1 - 27.9iT - 841T^{2} \) |
| 31 | \( 1 - 16.5T + 961T^{2} \) |
| 37 | \( 1 + 22.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 52.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 15.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 87.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 74.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 26.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 47.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 79.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 74.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 64.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 151. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 151. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 127.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 88.1T + 9.40e3T^{2} \) |
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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
−13.98306564892525200619889894491, −12.68248534524379914818157335319, −11.98286433469451224187911781574, −10.64085710969670679365573741532, −9.217890370223136611339567764820, −8.567547482270713749793131683889, −6.98192849639919910760406220800, −5.67502421642175896937537551461, −5.14870695169527684175194538135, −2.61190160907450069282470983728,
0.74216746192361817759719430994, 3.08786500788508493047673627415, 4.64909371019535636110849845373, 6.02668994974404233565411216692, 7.51541292818032936690731486175, 8.841416673624048784511384698517, 10.32032393638738840618529424885, 10.76140412898368563471635250371, 11.75786337273108578162739377255, 13.30446293079131128878768988015
