L(s) = 1 | + (−1.06 − 0.933i)2-s + (0.256 + 1.98i)4-s + (1.24 − 1.86i)5-s − 1.58·7-s + (1.57 − 2.34i)8-s + (−3.05 + 0.817i)10-s + (−3.92 + 3.92i)11-s + (−3.10 + 3.10i)13-s + (1.68 + 1.48i)14-s + (−3.86 + 1.01i)16-s − 1.48i·17-s + (4.94 + 4.94i)19-s + (4.00 + 1.98i)20-s + (7.82 − 0.504i)22-s − 6.61·23-s + ⋯ |
L(s) = 1 | + (−0.751 − 0.660i)2-s + (0.128 + 0.991i)4-s + (0.554 − 0.831i)5-s − 0.600·7-s + (0.558 − 0.829i)8-s + (−0.966 + 0.258i)10-s + (−1.18 + 1.18i)11-s + (−0.861 + 0.861i)13-s + (0.451 + 0.396i)14-s + (−0.967 + 0.254i)16-s − 0.359i·17-s + (1.13 + 1.13i)19-s + (0.896 + 0.443i)20-s + (1.66 − 0.107i)22-s − 1.38·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0618 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.249174 + 0.265088i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.249174 + 0.265088i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.06 + 0.933i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.24 + 1.86i)T \) |
good | 7 | \( 1 + 1.58T + 7T^{2} \) |
| 11 | \( 1 + (3.92 - 3.92i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.10 - 3.10i)T - 13iT^{2} \) |
| 17 | \( 1 + 1.48iT - 17T^{2} \) |
| 19 | \( 1 + (-4.94 - 4.94i)T + 19iT^{2} \) |
| 23 | \( 1 + 6.61T + 23T^{2} \) |
| 29 | \( 1 + (4.42 + 4.42i)T + 29iT^{2} \) |
| 31 | \( 1 + 1.50T + 31T^{2} \) |
| 37 | \( 1 + (-2.14 - 2.14i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.84iT - 41T^{2} \) |
| 43 | \( 1 + (0.322 + 0.322i)T + 43iT^{2} \) |
| 47 | \( 1 - 13.3iT - 47T^{2} \) |
| 53 | \( 1 + (0.931 + 0.931i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.14 - 1.14i)T - 59iT^{2} \) |
| 61 | \( 1 + (-2.67 - 2.67i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5.43 + 5.43i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.26iT - 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 79 | \( 1 + 6.52T + 79T^{2} \) |
| 83 | \( 1 + (0.973 - 0.973i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.83iT - 89T^{2} \) |
| 97 | \( 1 - 15.8iT - 97T^{2} \) |
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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
−10.21496541014489590386626779961, −9.660126808446658630694710653824, −9.437755538820857003010962259800, −7.974211161280036005347992018122, −7.58093348323179732121222110019, −6.30209098133381115498220224893, −5.06695903593169195986508055616, −4.11755271968867423029171250309, −2.63276820527477004783406448703, −1.69606866598851448405234871392,
0.22441323483238569397783583135, 2.31544364344507421590739307958, 3.30740421262839442999256711939, 5.36493347819214536671059915531, 5.68225260362304091407684746705, 6.83488133644485040794916286975, 7.52243711548601253191963241719, 8.376589028924312122893439097082, 9.446315281331484263701511968078, 10.09861978151751904465831642628