| L(s) = 1 | + 1.41i·2-s − 2.00·4-s − 4·7-s − 2.82i·8-s − 11.3i·11-s + 18·13-s − 5.65i·14-s + 4.00·16-s + 1.41i·17-s + 24·19-s + 16.0·22-s + 39.5i·23-s + 25.4i·26-s + 8.00·28-s − 38.1i·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 0.571·7-s − 0.353i·8-s − 1.02i·11-s + 1.38·13-s − 0.404i·14-s + 0.250·16-s + 0.0831i·17-s + 1.26·19-s + 0.727·22-s + 1.72i·23-s + 0.979i·26-s + 0.285·28-s − 1.31i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.617650556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.617650556\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4T + 49T^{2} \) |
| 11 | \( 1 + 11.3iT - 121T^{2} \) |
| 13 | \( 1 - 18T + 169T^{2} \) |
| 17 | \( 1 - 1.41iT - 289T^{2} \) |
| 19 | \( 1 - 24T + 361T^{2} \) |
| 23 | \( 1 - 39.5iT - 529T^{2} \) |
| 29 | \( 1 + 38.1iT - 841T^{2} \) |
| 31 | \( 1 - 4T + 961T^{2} \) |
| 37 | \( 1 - 56T + 1.36e3T^{2} \) |
| 41 | \( 1 + 24.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 80T + 1.84e3T^{2} \) |
| 47 | \( 1 - 28.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 4.24iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 62.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 110T + 3.72e3T^{2} \) |
| 67 | \( 1 + 32T + 4.48e3T^{2} \) |
| 71 | \( 1 + 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 46T + 5.32e3T^{2} \) |
| 79 | \( 1 + 36T + 6.24e3T^{2} \) |
| 83 | \( 1 - 5.65iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 57.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 14T + 9.40e3T^{2} \) |
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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
−11.05911471795338420190503484606, −9.805788438053003505914599375215, −9.126493883799602819098283891383, −8.142298526002289970192637215390, −7.33040936178153298637179560579, −6.02850853799443993156627975139, −5.71606397328536232569676668185, −4.07158327363988806282788583782, −3.14187023941735518812609087728, −0.947653413572085349838215185105,
1.04969804066432428537491198664, 2.59749466311685052474553599979, 3.73432563504302152485440846344, 4.80151418707651568730771595489, 6.05248727635176297107477836096, 7.07179134660151382391339825068, 8.255237386908147828976697700201, 9.195078659814883246740338050096, 9.946040128698342078468240179949, 10.80307141943373576788978479105
