L(s) = 1 | + 3-s + i·7-s + 9-s − i·11-s − 13-s + i·17-s − i·19-s + i·21-s − i·23-s + 27-s + i·29-s − 31-s − i·33-s − 37-s − 39-s + ⋯ |
L(s) = 1 | + 3-s + i·7-s + 9-s − i·11-s − 13-s + i·17-s − i·19-s + i·21-s − i·23-s + 27-s + i·29-s − 31-s − i·33-s − 37-s − 39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.256569246 + 0.1012940218i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.256569246 + 0.1012940218i\) |
\(L(1)\) |
\(\approx\) |
\(1.300878948 + 0.06371947111i\) |
\(L(1)\) |
\(\approx\) |
\(1.300878948 + 0.06371947111i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 + iT \) |
| 83 | \( 1 \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 \) |
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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
−31.140493160461236874267680108334, −30.025176858272000245706580775770, −29.21587788564283151298322562228, −27.47722986896765903282357696084, −26.78822325481701119582064912752, −25.675865628711337013310212065480, −24.80600706963828876879014288813, −23.58860177470804919568349164453, −22.43052904290868607267417096772, −20.94100812527287843801662767478, −20.19605930642581084831330716711, −19.28822952619682017586149413603, −17.92865800278483745952376481558, −16.68179962022588176151403382052, −15.30639713007402057690491715424, −14.29968882987436340954179543175, −13.339234354818173448544041464309, −12.050038494582874104451291858470, −10.22122068429988564963900231625, −9.46407308624763364953057040016, −7.77279003615778496570699863342, −7.06949902679163590776709668165, −4.80988918647143802088166890276, −3.53070561316865122763017226892, −1.89529916722627262351051662045,
2.14188797151303214127298530970, 3.37915952862833272581648625561, 5.12493611766135929449494574875, 6.79411334932139178655355693864, 8.34462144976619229675821321962, 9.07086811199259908788522758982, 10.517107785521342553731410278748, 12.14272144358779081508855793667, 13.23533335827136056596492406287, 14.54597980318294571545935317393, 15.31503489860425835620765317178, 16.61817529622017202018017978584, 18.24224211947875785558586989120, 19.1722550781691026754372442464, 20.06282256025485464809963208367, 21.5373535950618278622856710911, 21.9845526225858333616046509213, 24.007900543114771189895370909705, 24.63875701269362329776613925180, 25.78260755812807073093095938132, 26.6912354980227638351639312292, 27.77254318457701260637761382705, 29.04888135712363536394778721722, 30.2121946371932238783921335298, 31.22211684378438150742387198463