| L(s) = 1 | + i·2-s − 4-s + (0.539 − 2.17i)5-s − 1.07i·7-s − i·8-s + (2.17 + 0.539i)10-s − 6.34·11-s − 3.41i·13-s + 1.07·14-s + 16-s + 5.41i·17-s − 19-s + (−0.539 + 2.17i)20-s − 6.34i·22-s + 6.34i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.241 − 0.970i)5-s − 0.407i·7-s − 0.353i·8-s + (0.686 + 0.170i)10-s − 1.91·11-s − 0.948i·13-s + 0.288·14-s + 0.250·16-s + 1.31i·17-s − 0.229·19-s + (−0.120 + 0.485i)20-s − 1.35i·22-s + 1.32i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3294850507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3294850507\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.539 + 2.17i)T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 1.07iT - 7T^{2} \) |
| 11 | \( 1 + 6.34T + 11T^{2} \) |
| 13 | \( 1 + 3.41iT - 13T^{2} \) |
| 17 | \( 1 - 5.41iT - 17T^{2} \) |
| 23 | \( 1 - 6.34iT - 23T^{2} \) |
| 29 | \( 1 + 0.340T + 29T^{2} \) |
| 31 | \( 1 - 1.07T + 31T^{2} \) |
| 37 | \( 1 - 3.41iT - 37T^{2} \) |
| 41 | \( 1 + 7.60T + 41T^{2} \) |
| 43 | \( 1 - 11.1iT - 43T^{2} \) |
| 47 | \( 1 - 6.34iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 0.738T + 59T^{2} \) |
| 61 | \( 1 + 2.68T + 61T^{2} \) |
| 67 | \( 1 - 2.83iT - 67T^{2} \) |
| 71 | \( 1 - 2.83T + 71T^{2} \) |
| 73 | \( 1 + 6.83iT - 73T^{2} \) |
| 79 | \( 1 - 1.07T + 79T^{2} \) |
| 83 | \( 1 - 0.894iT - 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 + 3.65iT - 97T^{2} \) |
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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
−9.677032529712557563673202580562, −8.670691663987296803452966326415, −7.85454469725372450049387134052, −7.76909705346316116099607958637, −6.34437325228248525893757697690, −5.54539252209782841888999062270, −5.05096850719533793167359295554, −4.07471786719990899263776790590, −2.89080992944744505144796786435, −1.38079421182201414411848402703,
0.12179180070479452303132482089, 2.26031643762083475158380131555, 2.54594996841195646948524675541, 3.66911177223457823826096905366, 4.88668386644369064893215520033, 5.50612968597535100727602125238, 6.65914886377827287956507054772, 7.34270975495614874318811892179, 8.310827957019065320787657143628, 9.075648640791609283388760184347
