| L(s) = 1 | + (1.20 + 0.212i)2-s + (0.517 − 0.616i)3-s + (−0.477 − 0.173i)4-s + (2.06 − 0.857i)5-s + (0.752 − 0.631i)6-s + (−3.28 + 1.89i)7-s + (−2.65 − 1.53i)8-s + (0.408 + 2.31i)9-s + (2.66 − 0.593i)10-s + (0.618 − 1.07i)11-s + (−0.353 + 0.204i)12-s + (2.22 + 2.64i)13-s + (−4.35 + 1.58i)14-s + (0.539 − 1.71i)15-s + (−2.08 − 1.75i)16-s + (−2.96 − 0.522i)17-s + ⋯ |
| L(s) = 1 | + (0.850 + 0.149i)2-s + (0.298 − 0.355i)3-s + (−0.238 − 0.0868i)4-s + (0.923 − 0.383i)5-s + (0.307 − 0.257i)6-s + (−1.24 + 0.716i)7-s + (−0.937 − 0.541i)8-s + (0.136 + 0.772i)9-s + (0.843 − 0.187i)10-s + (0.186 − 0.323i)11-s + (−0.102 + 0.0589i)12-s + (0.616 + 0.734i)13-s + (−1.16 + 0.423i)14-s + (0.139 − 0.443i)15-s + (−0.522 − 0.438i)16-s + (−0.718 − 0.126i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43106 - 0.110737i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43106 - 0.110737i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.06 + 0.857i)T \) |
| 19 | \( 1 + (4.28 + 0.793i)T \) |
| good | 2 | \( 1 + (-1.20 - 0.212i)T + (1.87 + 0.684i)T^{2} \) |
| 3 | \( 1 + (-0.517 + 0.616i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (3.28 - 1.89i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.618 + 1.07i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.22 - 2.64i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.96 + 0.522i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.10 + 5.77i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.744 - 4.22i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.55 + 4.41i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 9.13iT - 37T^{2} \) |
| 41 | \( 1 + (-4.08 - 3.42i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.12 - 8.57i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-7.19 + 1.26i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (1.13 - 3.13i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.141 + 0.804i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (6.01 + 2.18i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.995 - 0.175i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (12.8 - 4.67i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-7.11 + 8.47i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.06 - 0.889i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (2.18 - 1.26i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.06 - 1.73i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (3.01 + 0.531i)T + (91.1 + 33.1i)T^{2} \) |
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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
−13.78389339362155878313703625615, −12.96558491775284158504813189501, −12.60859903938745530341193498611, −10.76839982290686887056258143202, −9.277374485833200937089781234008, −8.813690366806223688913503656411, −6.60819372563672806409462371784, −5.87404679333380363921720981004, −4.43857670263145102710728498298, −2.56941927728426093442666991826,
3.08916243635281795389953222876, 4.07397152468920828210157550610, 5.83332122351041303587087010490, 6.78981378585480113519709697467, 8.850586318747257896129783206630, 9.698834573518325536598792585154, 10.67841223938001807611506235997, 12.34432396725861807432324198366, 13.23208303310958763173447913516, 13.73284269366884100562475917746
