| L(s) = 1 | + (1.28 + 0.583i)2-s + (1.41 − 0.661i)3-s + (1.31 + 1.50i)4-s + (0.799 − 1.14i)5-s + (2.21 − 0.0242i)6-s + (−0.0847 + 0.146i)7-s + (0.821 + 2.70i)8-s + (−0.351 + 0.418i)9-s + (1.69 − 1.00i)10-s + (−2.03 − 0.546i)11-s + (2.86 + 1.26i)12-s + (−4.86 − 2.26i)13-s + (−0.194 + 0.139i)14-s + (0.379 − 2.15i)15-s + (−0.521 + 3.96i)16-s + (0.223 − 0.187i)17-s + ⋯ |
| L(s) = 1 | + (0.910 + 0.412i)2-s + (0.819 − 0.382i)3-s + (0.659 + 0.751i)4-s + (0.357 − 0.510i)5-s + (0.904 − 0.00989i)6-s + (−0.0320 + 0.0554i)7-s + (0.290 + 0.956i)8-s + (−0.117 + 0.139i)9-s + (0.536 − 0.317i)10-s + (−0.614 − 0.164i)11-s + (0.827 + 0.364i)12-s + (−1.35 − 0.629i)13-s + (−0.0520 + 0.0373i)14-s + (0.0979 − 0.555i)15-s + (−0.130 + 0.991i)16-s + (0.0543 − 0.0455i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.62980 + 0.375421i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.62980 + 0.375421i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.28 - 0.583i)T \) |
| 19 | \( 1 + (-2.41 + 3.62i)T \) |
| good | 3 | \( 1 + (-1.41 + 0.661i)T + (1.92 - 2.29i)T^{2} \) |
| 5 | \( 1 + (-0.799 + 1.14i)T + (-1.71 - 4.69i)T^{2} \) |
| 7 | \( 1 + (0.0847 - 0.146i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.03 + 0.546i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (4.86 + 2.26i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-0.223 + 0.187i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.157 - 0.893i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.126 + 1.44i)T + (-28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (-1.76 + 3.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.24 - 5.24i)T - 37iT^{2} \) |
| 41 | \( 1 + (0.800 + 0.291i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.02 + 0.714i)T + (14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-6.20 + 7.39i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (3.21 + 4.59i)T + (-18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (-0.672 - 0.0588i)T + (58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (6.42 + 9.17i)T + (-20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (-11.5 + 1.01i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-4.43 + 0.781i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (4.96 - 13.6i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (4.57 + 1.66i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-9.41 + 2.52i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (1.39 - 0.508i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (5.42 + 6.46i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−12.12824876783686519822271842548, −11.04436203081192851049844644280, −9.765222639644986683492843545482, −8.628171324405593575788574366782, −7.78456480275152274251464289008, −7.04252837295040008360313800705, −5.51244033903084062901995998934, −4.90733631315974109455173280491, −3.18933802769617691676927248447, −2.27039223669574016804691613479,
2.24300050220781053446061162444, 3.12509915968502991638255998563, 4.33013810995906887382452966139, 5.49134335343197479456678486279, 6.68057820666865976759573595095, 7.69626313104198039703846632019, 9.143417430702518733710079503531, 10.00227047857853947465767786879, 10.60842498167315701266656211681, 11.93305076798514104209047125921
