L(s) = 1 | + (−0.989 + 0.142i)7-s + (0.909 + 0.415i)11-s + (−0.142 + 0.989i)13-s + (−0.755 + 0.654i)17-s + (0.755 + 0.654i)19-s + (0.755 − 0.654i)29-s + (−0.959 − 0.281i)31-s + (0.841 − 0.540i)37-s + (0.841 + 0.540i)41-s + (0.959 − 0.281i)43-s − i·47-s + (0.959 − 0.281i)49-s + (−0.142 − 0.989i)53-s + (0.989 + 0.142i)59-s + (0.281 − 0.959i)61-s + ⋯ |
L(s) = 1 | + (−0.989 + 0.142i)7-s + (0.909 + 0.415i)11-s + (−0.142 + 0.989i)13-s + (−0.755 + 0.654i)17-s + (0.755 + 0.654i)19-s + (0.755 − 0.654i)29-s + (−0.959 − 0.281i)31-s + (0.841 − 0.540i)37-s + (0.841 + 0.540i)41-s + (0.959 − 0.281i)43-s − i·47-s + (0.959 − 0.281i)49-s + (−0.142 − 0.989i)53-s + (0.989 + 0.142i)59-s + (0.281 − 0.959i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.375360401 + 0.7395809613i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.375360401 + 0.7395809613i\) |
\(L(1)\) |
\(\approx\) |
\(1.008615583 + 0.1572099921i\) |
\(L(1)\) |
\(\approx\) |
\(1.008615583 + 0.1572099921i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 7 | \( 1 + (-0.989 + 0.142i)T \) |
| 11 | \( 1 + (0.909 + 0.415i)T \) |
| 13 | \( 1 + (-0.142 + 0.989i)T \) |
| 17 | \( 1 + (-0.755 + 0.654i)T \) |
| 19 | \( 1 + (0.755 + 0.654i)T \) |
| 29 | \( 1 + (0.755 - 0.654i)T \) |
| 31 | \( 1 + (-0.959 - 0.281i)T \) |
| 37 | \( 1 + (0.841 - 0.540i)T \) |
| 41 | \( 1 + (0.841 + 0.540i)T \) |
| 43 | \( 1 + (0.959 - 0.281i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.142 - 0.989i)T \) |
| 59 | \( 1 + (0.989 + 0.142i)T \) |
| 61 | \( 1 + (0.281 - 0.959i)T \) |
| 67 | \( 1 + (-0.415 - 0.909i)T \) |
| 71 | \( 1 + (0.415 + 0.909i)T \) |
| 73 | \( 1 + (-0.755 - 0.654i)T \) |
| 79 | \( 1 + (0.142 - 0.989i)T \) |
| 83 | \( 1 + (-0.841 + 0.540i)T \) |
| 89 | \( 1 + (0.959 - 0.281i)T \) |
| 97 | \( 1 + (-0.540 + 0.841i)T \) |
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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
−17.763712311251138402509242407234, −17.15537701843018175688001471984, −16.22960960527165922078059934013, −15.980169281959861664171076543341, −15.224814510673889048656429015109, −14.3625478515053487300123902424, −13.82642207529658178136851656756, −13.04652374339745748527966101882, −12.602255812705966417043207828924, −11.76519043805024090019187168785, −11.08875445505791002604254419650, −10.44076773505552747819407656110, −9.56386984321869218767691760058, −9.16462989140596075766901752975, −8.45425911572640019032276737450, −7.366765645356500168626329131048, −7.01366627077731770773834975164, −6.12028633983893317507866053143, −5.5892779902496180095477718034, −4.61160601952069432600630609388, −3.90148523811159477673067008897, −2.96379400390836577886475744793, −2.68364944870095674901648137049, −1.25141747694572157377597848827, −0.56061065870574539666325549185,
0.79568433162371149572489666587, 1.87941124104439837881204764177, 2.453588251789907089536103803324, 3.63566672651398842759581203554, 3.970397369366241106051682455318, 4.83579164951858592194549768030, 5.90149868062986256286295576291, 6.372759908060267582239679610385, 7.01528421089210672291822486143, 7.74345064715514805831803245933, 8.74548500491697388468828240059, 9.318462821991579315722434215422, 9.77112997944076415688479652155, 10.56478647725670130467505442921, 11.49696277359498923642340509890, 11.9319555860108555430207832301, 12.73299539345612774872051821239, 13.22186071411134199434028416676, 14.13705371921698428402916960286, 14.579304966532990090008957200343, 15.372050225786464363159781082804, 16.153757442986877389760104567655, 16.521639591507225684973360143244, 17.29297212954274486055558351083, 17.92458847574561097717292337288