| L(s) = 1 | + (−1.31 + 0.531i)2-s + (0.664 − 0.791i)3-s + (1.43 − 1.39i)4-s + (0.846 + 2.06i)5-s + (−0.450 + 1.39i)6-s + (1.98 + 3.43i)7-s + (−1.14 + 2.58i)8-s + (0.335 + 1.90i)9-s + (−2.20 − 2.26i)10-s + (−3.95 − 2.28i)11-s + (−0.149 − 2.06i)12-s + (−1.59 + 1.34i)13-s + (−4.42 − 3.44i)14-s + (2.20 + 0.704i)15-s + (0.120 − 3.99i)16-s + (−3.02 − 0.532i)17-s + ⋯ |
| L(s) = 1 | + (−0.926 + 0.375i)2-s + (0.383 − 0.457i)3-s + (0.717 − 0.696i)4-s + (0.378 + 0.925i)5-s + (−0.183 + 0.567i)6-s + (0.749 + 1.29i)7-s + (−0.403 + 0.915i)8-s + (0.111 + 0.634i)9-s + (−0.698 − 0.715i)10-s + (−1.19 − 0.689i)11-s + (−0.0430 − 0.595i)12-s + (−0.442 + 0.371i)13-s + (−1.18 − 0.921i)14-s + (0.568 + 0.182i)15-s + (0.0300 − 0.999i)16-s + (−0.732 − 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0488 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0488 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.738487 + 0.703257i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.738487 + 0.703257i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.31 - 0.531i)T \) |
| 5 | \( 1 + (-0.846 - 2.06i)T \) |
| 19 | \( 1 + (-2.31 - 3.69i)T \) |
| good | 3 | \( 1 + (-0.664 + 0.791i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-1.98 - 3.43i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.95 + 2.28i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.59 - 1.34i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.02 + 0.532i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (4.89 + 1.78i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-8.07 + 1.42i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.128 - 0.223i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.87T + 37T^{2} \) |
| 41 | \( 1 + (-3.49 + 4.16i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.50 - 1.27i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.32 + 7.54i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-5.94 - 2.16i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.0505 + 0.286i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-12.9 - 4.72i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-6.61 + 1.16i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.32 + 1.57i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.91 + 5.85i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.37 - 1.15i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.10 - 5.37i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.410 - 0.489i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-3.18 + 18.0i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.35252356294817540997582183865, −10.53683814707334878845460515898, −9.750502743587796091421499486511, −8.433286573275931460393891252201, −8.110850065437173395449961624129, −7.07321088259954921900316177513, −5.97777585818470848949240199790, −5.16322507279029755243153481288, −2.59473485709930834146725866197, −2.12716131801367150974809516498,
0.882371924769916065371463775211, 2.47798266082917076085350899764, 4.06922674573588023352950816860, 4.94780581061117866232601654232, 6.69822189764646896892849235608, 7.77828307600962297235325350811, 8.363041072277967109948974277458, 9.548422509099917402536675351071, 10.00591114700846120127379623430, 10.82837483020900148080701307332
