L(s) = 1 | + (0.531 + 0.919i)2-s + (1.35 − 2.35i)3-s + (0.435 − 0.754i)4-s + (−0.5 − 0.866i)5-s + 2.88·6-s + (0.964 + 2.46i)7-s + 3.05·8-s + (−2.18 − 3.78i)9-s + (0.531 − 0.919i)10-s + (0.5 − 0.866i)11-s + (−1.18 − 2.04i)12-s − 2.49·13-s + (−1.75 + 2.19i)14-s − 2.71·15-s + (0.748 + 1.29i)16-s + (−2.74 + 4.75i)17-s + ⋯ |
L(s) = 1 | + (0.375 + 0.650i)2-s + (0.783 − 1.35i)3-s + (0.217 − 0.377i)4-s + (−0.223 − 0.387i)5-s + 1.17·6-s + (0.364 + 0.931i)7-s + 1.07·8-s + (−0.728 − 1.26i)9-s + (0.167 − 0.290i)10-s + (0.150 − 0.261i)11-s + (−0.341 − 0.591i)12-s − 0.690·13-s + (−0.468 + 0.586i)14-s − 0.700·15-s + (0.187 + 0.324i)16-s + (−0.666 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.08675 - 0.806491i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.08675 - 0.806491i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.964 - 2.46i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.531 - 0.919i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.35 + 2.35i)T + (-1.5 - 2.59i)T^{2} \) |
| 13 | \( 1 + 2.49T + 13T^{2} \) |
| 17 | \( 1 + (2.74 - 4.75i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.108 - 0.187i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.17 + 3.77i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4.41T + 29T^{2} \) |
| 31 | \( 1 + (2.01 - 3.48i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.495 + 0.857i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.68T + 43T^{2} \) |
| 47 | \( 1 + (-6.02 - 10.4i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.217 - 0.376i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.02 + 5.23i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.54 - 4.41i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.192 - 0.334i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + (5.72 - 9.91i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.67 - 8.09i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 7.72T + 83T^{2} \) |
| 89 | \( 1 + (7.97 + 13.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.56T + 97T^{2} \) |
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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
−11.50758176052411993444160915463, −10.32368553508135821498662043496, −8.875097910692999061655466159079, −8.310002874819223178468687306994, −7.42032708123408911432961259449, −6.52142510006929911371774804005, −5.71166496060267287840333558924, −4.45744521363452346027745163068, −2.56352324491908032978328342710, −1.53238486595744688945418933195,
2.34814984300970183947112408301, 3.41241766818228772336186393546, 4.21570704575483437936498696585, 4.93494957297304082848990954062, 7.04108493599430770240848004765, 7.71513401682286095069666311966, 8.864464096810352650476145021145, 9.946692083313381206414266126272, 10.45977960804369728925979929709, 11.35927343599714372805751311995