L(s) = 1 | + (−2.26 + 1.69i)3-s + (2.20 + 0.358i)5-s + (0.137 − 1.92i)7-s + (1.40 − 4.78i)9-s + (−2.53 + 3.95i)11-s + (4.62 − 0.330i)13-s + (−5.60 + 2.92i)15-s + (1.23 − 3.32i)17-s + (3.07 − 6.73i)19-s + (2.94 + 4.58i)21-s + (2.02 − 4.34i)23-s + (4.74 + 1.58i)25-s + (1.96 + 5.27i)27-s + (−4.72 + 2.15i)29-s + (0.224 + 1.56i)31-s + ⋯ |
L(s) = 1 | + (−1.30 + 0.977i)3-s + (0.987 + 0.160i)5-s + (0.0520 − 0.727i)7-s + (0.468 − 1.59i)9-s + (−0.765 + 1.19i)11-s + (1.28 − 0.0917i)13-s + (−1.44 + 0.756i)15-s + (0.300 − 0.805i)17-s + (0.705 − 1.54i)19-s + (0.643 + 1.00i)21-s + (0.422 − 0.906i)23-s + (0.948 + 0.316i)25-s + (0.378 + 1.01i)27-s + (−0.876 + 0.400i)29-s + (0.0403 + 0.280i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19667 + 0.342141i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19667 + 0.342141i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.20 - 0.358i)T \) |
| 23 | \( 1 + (-2.02 + 4.34i)T \) |
good | 3 | \( 1 + (2.26 - 1.69i)T + (0.845 - 2.87i)T^{2} \) |
| 7 | \( 1 + (-0.137 + 1.92i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (2.53 - 3.95i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-4.62 + 0.330i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (-1.23 + 3.32i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (-3.07 + 6.73i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (4.72 - 2.15i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.224 - 1.56i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.83 - 3.36i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (4.89 - 1.43i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-7.30 - 9.76i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (-8.52 + 8.52i)T - 47iT^{2} \) |
| 53 | \( 1 + (6.22 + 0.445i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (-6.53 - 5.66i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-7.32 + 1.05i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-7.24 - 1.57i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (5.94 - 3.82i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-4.22 + 1.57i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (10.5 - 12.2i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (5.33 - 2.91i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (0.876 - 6.09i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (3.19 + 1.74i)T + (52.4 + 81.6i)T^{2} \) |
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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
−10.12036775034760279134750557691, −9.745158287430514746983969782462, −8.795012622183599216089798066396, −7.24681787652302129961774978395, −6.66848104954973480990798943559, −5.56318865423928385045487526379, −5.02087516019120385124247904443, −4.19038410305076568968087473107, −2.76520960746279785106678619335, −0.951882798515722959334756066559,
1.04160053138585595681012196144, 2.02516657649952244243994631912, 3.57583728591537304060232638978, 5.41394999159305304436173601290, 5.83725563359732432995346809583, 6.04240381729199407598313397708, 7.37976101302461121933699793997, 8.256753263146022012694400038415, 9.108979109659613661101409418865, 10.27860616727030212391134051333