L(s) = 1 | + (−0.173 + 0.984i)2-s + (0.0872 + 0.0731i)3-s + (−0.939 − 0.342i)4-s + (3.51 − 1.28i)5-s + (−0.0872 + 0.0731i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.518 − 2.94i)9-s + (0.649 + 3.68i)10-s + (−0.222 + 0.385i)11-s + (−0.0569 − 0.0986i)12-s + (3.82 − 3.21i)13-s + (0.939 − 0.342i)14-s + (0.400 + 0.145i)15-s + (0.766 + 0.642i)16-s + (−1.09 + 6.22i)17-s + ⋯ |
L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.0503 + 0.0422i)3-s + (−0.469 − 0.171i)4-s + (1.57 − 0.572i)5-s + (−0.0356 + 0.0298i)6-s + (−0.188 − 0.327i)7-s + (0.176 − 0.306i)8-s + (−0.172 − 0.980i)9-s + (0.205 + 1.16i)10-s + (−0.0671 + 0.116i)11-s + (−0.0164 − 0.0284i)12-s + (1.06 − 0.890i)13-s + (0.251 − 0.0914i)14-s + (0.103 + 0.0376i)15-s + (0.191 + 0.160i)16-s + (−0.266 + 1.51i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41836 + 0.161659i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41836 + 0.161659i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (4.23 - 1.04i)T \) |
good | 3 | \( 1 + (-0.0872 - 0.0731i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (-3.51 + 1.28i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (0.222 - 0.385i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.82 + 3.21i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.09 - 6.22i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.06 - 0.386i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.854 - 4.84i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.80 - 8.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 + (3.55 + 2.98i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (10.0 - 3.64i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (2.01 + 11.4i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-6.48 - 2.36i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.296 - 1.68i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (8.36 + 3.04i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.19 - 6.79i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-8.57 + 3.11i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (7.02 + 5.89i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-2.20 - 1.85i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (3.11 + 5.40i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.26 + 3.58i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (2.94 - 16.6i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−12.37351919590053773939448495376, −10.51368623044824855844801891891, −10.12840682489670813485251296281, −8.782508728638790631616791122556, −8.524780813164322350526449925414, −6.62793216067761777904008763313, −6.13572521717514930524032315521, −5.10865356367553591353463357204, −3.55537130420103375800281286563, −1.43816881051598081947059620136,
1.95835369385258660403624956947, 2.79630123861634302083919616284, 4.64765875994218519041603551680, 5.86557210640020034313902324007, 6.77853372305817383410728846563, 8.345275337783495320041277527672, 9.309356082578460955095989407080, 10.01525966300396877570481082706, 10.97478949624503891924047235885, 11.62170175370626097764228588343