| L(s) = 1 | + 1.41·2-s + (2.16 + 2.07i)3-s + 2.00·4-s + (3.06 + 2.93i)6-s − 2.64i·7-s + 2.82·8-s + (0.400 + 8.99i)9-s + 21.5i·11-s + (4.33 + 4.14i)12-s − 15.4i·13-s − 3.74i·14-s + 4.00·16-s + 20.3·17-s + (0.566 + 12.7i)18-s + 5.86·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + (0.722 + 0.691i)3-s + 0.500·4-s + (0.511 + 0.488i)6-s − 0.377i·7-s + 0.353·8-s + (0.0445 + 0.999i)9-s + 1.96i·11-s + (0.361 + 0.345i)12-s − 1.19i·13-s − 0.267i·14-s + 0.250·16-s + 1.19·17-s + (0.0314 + 0.706i)18-s + 0.308·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.009972174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.009972174\) |
| \(L(2)\) |
|
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 | \( 1 + (-2.16 - 2.07i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 11 | \( 1 - 21.5iT - 121T^{2} \) |
| 13 | \( 1 + 15.4iT - 169T^{2} \) |
| 17 | \( 1 - 20.3T + 289T^{2} \) |
| 19 | \( 1 - 5.86T + 361T^{2} \) |
| 23 | \( 1 - 11.3T + 529T^{2} \) |
| 29 | \( 1 - 3.43iT - 841T^{2} \) |
| 31 | \( 1 + 21.0T + 961T^{2} \) |
| 37 | \( 1 - 58.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 36.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 50.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 36.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 40.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 110. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 41.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 109. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 14.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 19.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 122.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 74.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 92.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 5.71iT - 9.40e3T^{2} \) |
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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
−9.967861417926500308764763980235, −9.320997319240615966350190552871, −7.77916728310233361446510146790, −7.73851458649336098108180947019, −6.49136798736556451103894583102, −5.12213833383092615863239271209, −4.75927875563956141657839688599, −3.60156006704783012558581595171, −2.86081606487341373196999015835, −1.57892128804559552159423311633,
0.938492583462670517538972895917, 2.23062459383345896505418944184, 3.26947535187269317082453545447, 3.91335539579816417897917383079, 5.49070886278171354693519123337, 6.01879195640985843003787715011, 7.04307631973494734884357146099, 7.76930919557984557188981409806, 8.810201362032875150574676057269, 9.177276206886596385677738246278
