L(s) = 1 | − i·2-s + i·3-s − 4-s + i·5-s + 6-s + i·7-s + i·8-s − 9-s + 10-s − 11-s − i·12-s − 13-s + 14-s − 15-s + 16-s + i·17-s + ⋯ |
L(s) = 1 | − i·2-s + i·3-s − 4-s + i·5-s + 6-s + i·7-s + i·8-s − 9-s + 10-s − 11-s − i·12-s − 13-s + 14-s − 15-s + 16-s + i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1013 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1013 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.3167617737 + 0.4684139638i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.3167617737 + 0.4684139638i\) |
\(L(1)\) |
\(\approx\) |
\(0.6418234002 + 0.2742444674i\) |
\(L(1)\) |
\(\approx\) |
\(0.6418234002 + 0.2742444674i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1013 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + iT \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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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
−20.53691416721655920862495049018, −20.2191453416265197447153752608, −19.14315597223840532144332838548, −18.38062442061859792538638222864, −17.54589556255918472007090416588, −17.04762939618379349305154538190, −16.29684308603271843684157689736, −15.55348470187681742217712921479, −14.37676910091854627313437405197, −13.543927168763260205418327833989, −13.360641337872018166921370907845, −12.3302607568908354123230774346, −11.61311997614761389609799631210, −10.06079730345565285686395327945, −9.48970248909942719442213598476, −8.26824048689304697450781568094, −7.71876748740764844673564656110, −7.22889765357641779376899970138, −6.13200512015272238131442676114, −5.2002557403493387445932495698, −4.63525237141386222104381754090, −3.32352752283224781101198756531, −1.91839117239393032061784371390, −0.55550527540525527285848700697, −0.20395277096989853895183627700,
1.86809081812340334748445082587, 2.84266728735015196446817108287, 3.22045084629321045029560121089, 4.4333595484162157010386792251, 5.32059159453269498932001952444, 5.97584574940795130328902777656, 7.57739913994507764556338089173, 8.457886807340262472586164880648, 9.37270227879199138409450716837, 10.160957580855436805365903224108, 10.56622232429154092642752309334, 11.52888411191200191073780513738, 12.1412687359818171635238532542, 13.116624761440457869445736446781, 14.3671499355635926461476836816, 14.627394213343216595100257454360, 15.56891344548353549512743681547, 16.3568433416663638552717750431, 17.79811909626960581446680674095, 17.931702820267497854286973004914, 19.01968801832545274831376142138, 19.66358775318708461641556096637, 20.46253178767999701560474325592, 21.548601684392177870023471110224, 21.73038422424848832326591745081