| L(s) = 1 | + (−0.654 − 0.755i)3-s + (−0.000386 − 0.00269i)5-s + (−0.858 + 1.87i)7-s + (−0.142 + 0.989i)9-s + (2.08 + 0.611i)11-s + (0.103 + 0.227i)13-s + (−0.00178 + 0.00205i)15-s + (5.66 + 3.63i)17-s + (−0.372 + 0.239i)19-s + (1.98 − 0.582i)21-s + (−3.86 + 2.84i)23-s + (4.79 − 1.40i)25-s + (0.841 − 0.540i)27-s + (6.42 + 4.13i)29-s + (0.394 − 0.455i)31-s + ⋯ |
| L(s) = 1 | + (−0.378 − 0.436i)3-s + (−0.000173 − 0.00120i)5-s + (−0.324 + 0.710i)7-s + (−0.0474 + 0.329i)9-s + (0.627 + 0.184i)11-s + (0.0288 + 0.0630i)13-s + (−0.000459 + 0.000530i)15-s + (1.37 + 0.882i)17-s + (−0.0853 + 0.0548i)19-s + (0.432 − 0.127i)21-s + (−0.805 + 0.593i)23-s + (0.959 − 0.281i)25-s + (0.161 − 0.104i)27-s + (1.19 + 0.767i)29-s + (0.0708 − 0.0818i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22564 + 0.292851i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22564 + 0.292851i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (3.86 - 2.84i)T \) |
| good | 5 | \( 1 + (0.000386 + 0.00269i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (0.858 - 1.87i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-2.08 - 0.611i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.103 - 0.227i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-5.66 - 3.63i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.372 - 0.239i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-6.42 - 4.13i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.394 + 0.455i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.38 + 9.62i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.633 - 4.40i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-1.62 - 1.87i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 1.92T + 47T^{2} \) |
| 53 | \( 1 + (1.18 - 2.59i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.80 - 3.94i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.84 + 2.12i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-0.318 + 0.0934i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (10.8 - 3.17i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (10.3 - 6.64i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (2.58 + 5.66i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.103 + 0.718i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-6.45 - 7.45i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (1.56 + 10.8i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.85411179385549202666659837657, −10.02240511465111647680012520094, −9.063149859804526296450741866066, −8.180863770117343832737194396927, −7.19184914526916699121567573506, −6.17468757659412876851878472376, −5.54854011369313264939958229857, −4.19141085069644185935953314585, −2.87649453978398655474967563977, −1.38613373518032619054393092901,
0.897198225972634949010847896090, 2.98653598011346161773594940353, 4.04956428291308445297401942202, 5.04150244690663629897993894226, 6.17671039965101631711445475085, 6.99069849912764187063617937680, 8.053371522188122263399767457550, 9.101610665224922592538439698475, 10.06678072474368302707725295490, 10.47373859398293368753624666072
