L(s) = 1 | − 1.41·2-s + 2.00·4-s + 2.23i·5-s − 11.8i·7-s − 2.82·8-s − 3.16i·10-s − 20.7i·11-s + 7.75·13-s + 16.7i·14-s + 4.00·16-s + 22.9i·17-s − 21.0i·19-s + 4.47i·20-s + 29.3i·22-s + (21.9 + 6.71i)23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.500·4-s + 0.447i·5-s − 1.68i·7-s − 0.353·8-s − 0.316i·10-s − 1.88i·11-s + 0.596·13-s + 1.19i·14-s + 0.250·16-s + 1.34i·17-s − 1.10i·19-s + 0.223i·20-s + 1.33i·22-s + (0.956 + 0.292i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.440890977\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.440890977\) |
\(L(2)\) |
|
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 | \( 1 + (-21.9 - 6.71i)T \) |
good | 7 | \( 1 + 11.8iT - 49T^{2} \) |
| 11 | \( 1 + 20.7iT - 121T^{2} \) |
| 13 | \( 1 - 7.75T + 169T^{2} \) |
| 17 | \( 1 - 22.9iT - 289T^{2} \) |
| 19 | \( 1 + 21.0iT - 361T^{2} \) |
| 29 | \( 1 - 38.2T + 841T^{2} \) |
| 31 | \( 1 - 33.3T + 961T^{2} \) |
| 37 | \( 1 + 49.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 0.679T + 1.68e3T^{2} \) |
| 43 | \( 1 - 7.99iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 85.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 66.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 110.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 16.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 117. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 28.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 31.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 105. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 43.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 49.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 69.6iT - 9.40e3T^{2} \) |
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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
−8.634094123530411314564893797857, −8.005573167848041308456455042633, −7.22114802544061482916361694364, −6.46071682313037912759112574167, −5.88257012552311644534874626808, −4.46084441816531580566177977000, −3.57555184280955158485295664558, −2.85663145586902031271908072287, −1.19203543565429720840188038744, −0.55970021844600723507728475626,
1.18664174978133481462560904008, 2.23079882969989444874537783202, 2.95393485840776362747179597334, 4.53623599656229171896867985365, 5.15493253214656036664491061722, 6.11902152692400116365692227141, 6.88894866879087311598362436932, 7.78078871282195162531569192493, 8.580361223044125377344657659887, 9.087255664281258043146470224237