| L(s) = 1 | + (−0.781 + 0.196i)2-s + (−0.476 − 0.630i)3-s + (−1.19 + 0.639i)4-s + (2.44 − 2.85i)5-s + (0.495 + 0.398i)6-s + (0.409 − 0.130i)7-s + (1.99 − 1.81i)8-s + (0.650 − 2.28i)9-s + (−1.34 + 2.70i)10-s + (1.70 + 1.76i)11-s + (0.970 + 0.446i)12-s + (−1.67 − 1.52i)13-s + (−0.294 + 0.182i)14-s + (−2.96 − 0.182i)15-s + (0.292 − 0.440i)16-s + (−3.58 + 2.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.552 + 0.138i)2-s + (−0.274 − 0.364i)3-s + (−0.595 + 0.319i)4-s + (1.09 − 1.27i)5-s + (0.202 + 0.162i)6-s + (0.154 − 0.0492i)7-s + (0.705 − 0.642i)8-s + (0.216 − 0.761i)9-s + (−0.426 + 0.856i)10-s + (0.514 + 0.530i)11-s + (0.280 + 0.128i)12-s + (−0.464 − 0.423i)13-s + (−0.0786 + 0.0486i)14-s + (−0.765 − 0.0471i)15-s + (0.0730 − 0.110i)16-s + (−0.869 + 0.699i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.687757 - 0.301035i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.687757 - 0.301035i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (9.96 + 1.93i)T \) |
| good | 2 | \( 1 + (0.781 - 0.196i)T + (1.76 - 0.946i)T^{2} \) |
| 3 | \( 1 + (0.476 + 0.630i)T + (-0.820 + 2.88i)T^{2} \) |
| 5 | \( 1 + (-2.44 + 2.85i)T + (-0.766 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-0.409 + 0.130i)T + (5.71 - 4.04i)T^{2} \) |
| 11 | \( 1 + (-1.70 - 1.76i)T + (-0.338 + 10.9i)T^{2} \) |
| 13 | \( 1 + (1.67 + 1.52i)T + (1.19 + 12.9i)T^{2} \) |
| 17 | \( 1 + (3.58 - 2.88i)T + (3.63 - 16.6i)T^{2} \) |
| 19 | \( 1 + (-0.838 - 0.103i)T + (18.4 + 4.63i)T^{2} \) |
| 23 | \( 1 + (-1.83 - 6.44i)T + (-19.5 + 12.1i)T^{2} \) |
| 29 | \( 1 + (0.433 + 1.23i)T + (-22.5 + 18.1i)T^{2} \) |
| 31 | \( 1 + (-1.19 - 2.39i)T + (-18.6 + 24.7i)T^{2} \) |
| 37 | \( 1 + (0.00704 + 0.0760i)T + (-36.3 + 6.79i)T^{2} \) |
| 41 | \( 1 + (-7.12 - 8.32i)T + (-6.28 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.928 - 0.427i)T + (27.9 - 32.6i)T^{2} \) |
| 47 | \( 1 + (-5.57 + 9.66i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.25 + 5.31i)T + (-36.8 + 38.0i)T^{2} \) |
| 59 | \( 1 + (-9.25 - 2.94i)T + (48.1 + 34.0i)T^{2} \) |
| 61 | \( 1 + (-10.7 - 4.16i)T + (45.0 + 41.0i)T^{2} \) |
| 67 | \( 1 + (0.807 + 3.68i)T + (-60.8 + 28.0i)T^{2} \) |
| 71 | \( 1 + (4.36 - 12.3i)T + (-55.3 - 44.5i)T^{2} \) |
| 73 | \( 1 + (14.0 - 2.62i)T + (68.0 - 26.3i)T^{2} \) |
| 79 | \( 1 + (15.0 + 2.80i)T + (73.6 + 28.5i)T^{2} \) |
| 83 | \( 1 + (2.45 - 11.2i)T + (-75.4 - 34.6i)T^{2} \) |
| 89 | \( 1 + (-4.96 + 3.07i)T + (39.6 - 79.6i)T^{2} \) |
| 97 | \( 1 + (-3.35 - 2.69i)T + (20.7 + 94.7i)T^{2} \) |
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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
−13.16896212191326952297305871377, −12.97005575759760970537309215085, −11.82617322789932777015469849088, −9.954172125038445025090353281200, −9.348590178536910154476594614010, −8.437548699365443646333757986877, −7.02262340079025735287197571126, −5.57072675250494150279179931423, −4.28216826749556950491933569075, −1.31978863582562915566826204131,
2.31204865004045070089544799659, 4.61213106199754888079528528746, 5.91984997242266931519622392408, 7.21630014144684996185502936941, 8.868286609709583695089290373293, 9.800180930864263112596389976517, 10.63382349708683193974013752863, 11.28245861113813823809091396691, 13.22780414196171826673758332585, 14.11143642754820175542800235174
