L(s) = 1 | + (0.940 − 0.339i)2-s + (0.768 − 0.639i)4-s + (−0.0611 − 0.998i)5-s + (0.917 + 0.396i)7-s + (0.505 − 0.862i)8-s + (−0.396 − 0.917i)10-s + (0.994 − 0.101i)11-s + (0.999 + 0.0407i)13-s + (0.998 + 0.0611i)14-s + (0.182 − 0.983i)16-s + (0.909 − 0.415i)17-s + (−0.639 − 0.768i)19-s + (−0.685 − 0.728i)20-s + (0.900 − 0.433i)22-s + (−0.992 + 0.122i)25-s + (0.953 − 0.301i)26-s + ⋯ |
L(s) = 1 | + (0.940 − 0.339i)2-s + (0.768 − 0.639i)4-s + (−0.0611 − 0.998i)5-s + (0.917 + 0.396i)7-s + (0.505 − 0.862i)8-s + (−0.396 − 0.917i)10-s + (0.994 − 0.101i)11-s + (0.999 + 0.0407i)13-s + (0.998 + 0.0611i)14-s + (0.182 − 0.983i)16-s + (0.909 − 0.415i)17-s + (−0.639 − 0.768i)19-s + (−0.685 − 0.728i)20-s + (0.900 − 0.433i)22-s + (−0.992 + 0.122i)25-s + (0.953 − 0.301i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.048257946 - 2.535099440i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.048257946 - 2.535099440i\) |
\(L(1)\) |
\(\approx\) |
\(2.072916649 - 0.9476150084i\) |
\(L(1)\) |
\(\approx\) |
\(2.072916649 - 0.9476150084i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.940 - 0.339i)T \) |
| 5 | \( 1 + (-0.0611 - 0.998i)T \) |
| 7 | \( 1 + (0.917 + 0.396i)T \) |
| 11 | \( 1 + (0.994 - 0.101i)T \) |
| 13 | \( 1 + (0.999 + 0.0407i)T \) |
| 17 | \( 1 + (0.909 - 0.415i)T \) |
| 19 | \( 1 + (-0.639 - 0.768i)T \) |
| 31 | \( 1 + (-0.0815 + 0.996i)T \) |
| 37 | \( 1 + (0.574 + 0.818i)T \) |
| 41 | \( 1 + (-0.281 - 0.959i)T \) |
| 43 | \( 1 + (0.0815 + 0.996i)T \) |
| 47 | \( 1 + (-0.781 + 0.623i)T \) |
| 53 | \( 1 + (-0.882 + 0.470i)T \) |
| 59 | \( 1 + (0.654 - 0.755i)T \) |
| 61 | \( 1 + (-0.830 + 0.557i)T \) |
| 67 | \( 1 + (-0.101 + 0.994i)T \) |
| 71 | \( 1 + (0.947 + 0.320i)T \) |
| 73 | \( 1 + (-0.202 + 0.979i)T \) |
| 79 | \( 1 + (-0.983 + 0.182i)T \) |
| 83 | \( 1 + (-0.742 - 0.670i)T \) |
| 89 | \( 1 + (-0.359 + 0.933i)T \) |
| 97 | \( 1 + (0.162 - 0.986i)T \) |
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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
−20.27798657115585545052945793200, −19.49772654260597921994079298096, −18.59323189343109422013282236983, −17.87999416622203766977247637712, −16.96870769704740664015319600360, −16.56158575901655954141567799793, −15.40461371569560224055660563161, −14.82690120183399386322885962119, −14.32785932493759867252354622949, −13.78336785976268410252367634888, −12.86192207167790033858090979332, −11.930128565304809088459894123348, −11.322221595476949564597118624486, −10.76694104328310356686004548842, −9.91111714733780945303785557570, −8.553926970779418100729409741704, −7.877874508554991081453772133078, −7.21019355192794928226797563578, −6.253136024285353179663825706954, −5.89246381304200432547175122340, −4.68608890939169166947122757020, −3.800369087093202587500575845430, −3.480058937361187874446898267903, −2.15345654202770265022167733710, −1.4379967338681221560140775317,
1.1650838345384017332984002534, 1.47886476243416440114037823951, 2.71115825079497328580761728605, 3.73303867939838529330544195632, 4.48297972393206793334163463521, 5.10730474807265134987204458025, 5.88998874837946989595717700433, 6.66547656678527612627879755464, 7.76973721295283823925005614116, 8.62037590579561418496953748236, 9.26900390035012050977501958434, 10.27667159374212141896253240806, 11.45231217328225101629686814772, 11.49755767413034267966361143331, 12.505831319505551318687701218288, 13.04585455976826891591464475728, 14.042929653856008438083799924215, 14.39896148544963743478158039595, 15.393671829769985011230276873205, 15.97207626618872754445573541269, 16.772992880399888988559918970643, 17.49530297485249521405277952154, 18.49504933805180519828656161115, 19.28040801368508234875247749484, 20.01469373660963222936767671988