L(s) = 1 | + (0.831 + 0.555i)2-s + (0.555 + 0.831i)3-s + (0.382 + 0.923i)4-s + (−0.980 − 0.195i)5-s + i·6-s + (−0.195 + 0.980i)8-s + (−0.382 + 0.923i)9-s + (−0.707 − 0.707i)10-s + (−0.555 + 0.831i)12-s + (−0.382 − 0.923i)15-s + (−0.707 + 0.707i)16-s + (−0.831 + 0.555i)17-s + (−0.831 + 0.555i)18-s + (0.707 − 0.292i)19-s + (−0.195 − 0.980i)20-s + ⋯ |
L(s) = 1 | + (0.831 + 0.555i)2-s + (0.555 + 0.831i)3-s + (0.382 + 0.923i)4-s + (−0.980 − 0.195i)5-s + i·6-s + (−0.195 + 0.980i)8-s + (−0.382 + 0.923i)9-s + (−0.707 − 0.707i)10-s + (−0.555 + 0.831i)12-s + (−0.382 − 0.923i)15-s + (−0.707 + 0.707i)16-s + (−0.831 + 0.555i)17-s + (−0.831 + 0.555i)18-s + (0.707 − 0.292i)19-s + (−0.195 − 0.980i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.598990445\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.598990445\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.831 - 0.555i)T \) |
| 3 | \( 1 + (-0.555 - 0.831i)T \) |
| 5 | \( 1 + (0.980 + 0.195i)T \) |
| 17 | \( 1 + (0.831 - 0.555i)T \) |
good | 7 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 11 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (-0.785 + 1.17i)T + (-0.382 - 0.923i)T^{2} \) |
| 29 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + (-1.63 + 1.08i)T + (0.382 - 0.923i)T^{2} \) |
| 37 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 41 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 43 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + (-0.275 + 0.275i)T - iT^{2} \) |
| 53 | \( 1 + (-0.425 - 1.02i)T + (-0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 73 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 79 | \( 1 + (0.324 + 0.216i)T + (0.382 + 0.923i)T^{2} \) |
| 83 | \( 1 + (0.636 + 1.53i)T + (-0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (0.923 + 0.382i)T^{2} \) |
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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
−10.60689526337316620396585205166, −9.358119416419886191232412797054, −8.526708574412993516098818965542, −8.005343783562708440794315858800, −7.10037224352414429964382687665, −6.09317279627518405377900994854, −4.80208579409195928110052882611, −4.44724761054085639217957645820, −3.44920010401175036984656407169, −2.56471305480360113165921321398,
1.22664354114791953041737218556, 2.70410768000419378180703777729, 3.37146976327067581398487489892, 4.40898767480524850426026504466, 5.46973551704015787589079570648, 6.70621551764730698058130446459, 7.14219218697451247827905996449, 8.137173748984588445107503826433, 9.053544395045903542663182384059, 9.962151020062601418927534252496