L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s − 10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 17-s + (−0.5 − 0.866i)20-s + (0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 26-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s − 10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 17-s + (−0.5 − 0.866i)20-s + (0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8891851025 - 0.3236369100i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8891851025 - 0.3236369100i\) |
\(L(1)\) |
\(\approx\) |
\(0.9030205325 + 0.3286725947i\) |
\(L(1)\) |
\(\approx\) |
\(0.9030205325 + 0.3286725947i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 + T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−28.02186028723251582399137892728, −26.61833463556177400859935764538, −25.336085684415408613952660779263, −24.30101594343430839750008213519, −23.32748597053188037026557947100, −22.65975369079663304650750374351, −21.28733047418820331671529263919, −20.832633469959521693231529587490, −19.64273322734451100853293020057, −18.92619380042927595259454669952, −17.89982728685744377189747764103, −16.31995156898386475833860747654, −15.498802257472227786697319795818, −14.35445662386233576589561027449, −13.0770492113864910745008970335, −12.29900291908717180768812373643, −11.67889602494808519306103436174, −10.16382230374122978200885689854, −9.24632814631317405998189008637, −8.19298024538805857764961641996, −6.35036750544842780389774707038, −5.13036796311919170328579476012, −4.184184350055567962583556254733, −2.80127303762705211794058000888, −1.4161510475773265526464407584,
0.302353214030493977628937531333, 3.18162076485516617842880190244, 3.70142986252540151326684144136, 5.39798074432414684999301822360, 6.462065742097177797197550152105, 7.51973345440327828903868580705, 8.27021680487133371488547745471, 9.984717256631673875531262668373, 11.034548184436482169926642132081, 12.356505354543266100683686503174, 13.54718586354016344418567204501, 14.18371041067894424942122236290, 15.451571442679535033538674979098, 16.07592517254436788173753629980, 17.17709014934068286873983456510, 18.28177669836494704561303702878, 19.21293107156440324427394849964, 20.53000025648083990946172480772, 21.68749104877824176686112308875, 22.6691915610629698957867511982, 23.33000574782244932067448316434, 24.04302373712285554637563280184, 25.47812291272055701105364605520, 26.07414210454181284143896918681, 26.93228696153651096999666799515