L(s) = 1 | + (1.68 − 4.70i)5-s + (−10.9 − 6.33i)7-s + (−16.1 − 9.35i)11-s + (5.88 − 3.39i)13-s + 11.5·17-s + 24.2·19-s + (−14.6 − 25.3i)23-s + (−19.3 − 15.8i)25-s + (12.9 + 7.47i)29-s + (−6.91 − 11.9i)31-s + (−48.2 + 40.9i)35-s − 38.2i·37-s + (26.7 − 15.4i)41-s + (−11.3 − 6.53i)43-s + (−2.84 + 4.92i)47-s + ⋯ |
L(s) = 1 | + (0.336 − 0.941i)5-s + (−1.56 − 0.904i)7-s + (−1.47 − 0.850i)11-s + (0.452 − 0.261i)13-s + 0.679·17-s + 1.27·19-s + (−0.635 − 1.10i)23-s + (−0.773 − 0.633i)25-s + (0.446 + 0.257i)29-s + (−0.223 − 0.386i)31-s + (−1.37 + 1.17i)35-s − 1.03i·37-s + (0.653 − 0.377i)41-s + (−0.263 − 0.152i)43-s + (−0.0605 + 0.104i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5794757029\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5794757029\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.68 + 4.70i)T \) |
good | 7 | \( 1 + (10.9 + 6.33i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (16.1 + 9.35i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-5.88 + 3.39i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 11.5T + 289T^{2} \) |
| 19 | \( 1 - 24.2T + 361T^{2} \) |
| 23 | \( 1 + (14.6 + 25.3i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-12.9 - 7.47i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (6.91 + 11.9i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 38.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-26.7 + 15.4i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (11.3 + 6.53i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (2.84 - 4.92i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 57.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + (26.2 - 15.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (38.5 - 66.7i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (111. - 64.1i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 35.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (70.0 - 121. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-59.4 + 102. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 75.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-73.1 - 42.2i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−8.762817215814727588151828050894, −7.903459028631049020480153287374, −7.22580372247605167992512590828, −5.97339341736151078285270199684, −5.68571567216076081145980532790, −4.50208944001892652959222737135, −3.47334818924216261357679753263, −2.72100773924842361503606531162, −0.976687284005537035847216189447, −0.17974674931647619707022602837,
1.87933437749156677564050351059, 3.05331444536193206300295963783, 3.26746587591824478073845624221, 4.94194941938873353680562884669, 5.83898539225827339328229768728, 6.33899738176146467788881086635, 7.34289492218483400924452949204, 7.903081907323877244500743351382, 9.214205893831876247697084588708, 9.817099114651289811561210612439