| L(s) = 1 | − 1.41·2-s − 1.73i·3-s + 2.00·4-s + 2.44i·6-s + (6.84 + 1.46i)7-s − 2.82·8-s − 2.99·9-s − 5.87·11-s − 3.46i·12-s + 2.06i·13-s + (−9.68 − 2.07i)14-s + 4.00·16-s + 2.59i·17-s + 4.24·18-s − 14.3i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.500·4-s + 0.408i·6-s + (0.977 + 0.209i)7-s − 0.353·8-s − 0.333·9-s − 0.534·11-s − 0.288i·12-s + 0.158i·13-s + (−0.691 − 0.148i)14-s + 0.250·16-s + 0.152i·17-s + 0.235·18-s − 0.757i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.147189680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.147189680\) |
| \(L(2)\) |
|
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 | \( 1 + (-6.84 - 1.46i)T \) |
| good | 11 | \( 1 + 5.87T + 121T^{2} \) |
| 13 | \( 1 - 2.06iT - 169T^{2} \) |
| 17 | \( 1 - 2.59iT - 289T^{2} \) |
| 19 | \( 1 + 14.3iT - 361T^{2} \) |
| 23 | \( 1 + 4.73T + 529T^{2} \) |
| 29 | \( 1 + 16.1T + 841T^{2} \) |
| 31 | \( 1 + 48.3iT - 961T^{2} \) |
| 37 | \( 1 - 46.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 74.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 10.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 35.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 39.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 60.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 73.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 39.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 39.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 11.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 41.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 92.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 4.26iT - 9.40e3T^{2} \) |
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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
−9.312934003026015937381718657330, −8.611543512852181556995816917543, −7.77265795482404879375518743189, −7.31779078329445589111743861257, −6.16370854914871283588738528784, −5.36198826645149010193989187867, −4.19216555184565603355791019013, −2.64469332724236390578048716460, −1.82596085724235917585483297040, −0.48567105583137830527059729474,
1.19645182395886082381967336403, 2.47192566438455107447146909126, 3.69317911808108334067284106790, 4.81532520319675440690920459422, 5.59227688849515734908099181968, 6.71868116568338847251457048205, 7.81257528199542904382836117540, 8.223274185454707824647061061015, 9.153944918852459696970842183424, 10.02860645950267290699711903780
