L(s) = 1 | + (−0.985 − 1.01i)2-s + (−0.0577 + 1.99i)4-s + 0.206·5-s − i·7-s + (2.08 − 1.91i)8-s + (−0.203 − 0.209i)10-s + 6.42i·11-s − 3.52i·13-s + (−1.01 + 0.985i)14-s + (−3.99 − 0.230i)16-s + 3.90i·17-s − 5.48·19-s + (−0.0119 + 0.413i)20-s + (6.51 − 6.32i)22-s − 0.880·23-s + ⋯ |
L(s) = 1 | + (−0.696 − 0.717i)2-s + (−0.0288 + 0.999i)4-s + 0.0925·5-s − 0.377i·7-s + (0.737 − 0.675i)8-s + (−0.0645 − 0.0663i)10-s + 1.93i·11-s − 0.977i·13-s + (−0.271 + 0.263i)14-s + (−0.998 − 0.0577i)16-s + 0.947i·17-s − 1.25·19-s + (−0.00267 + 0.0925i)20-s + (1.38 − 1.34i)22-s − 0.183·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1746277224\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1746277224\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.985 + 1.01i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 0.206T + 5T^{2} \) |
| 11 | \( 1 - 6.42iT - 11T^{2} \) |
| 13 | \( 1 + 3.52iT - 13T^{2} \) |
| 17 | \( 1 - 3.90iT - 17T^{2} \) |
| 19 | \( 1 + 5.48T + 19T^{2} \) |
| 23 | \( 1 + 0.880T + 23T^{2} \) |
| 29 | \( 1 + 3.20T + 29T^{2} \) |
| 31 | \( 1 + 0.0631iT - 31T^{2} \) |
| 37 | \( 1 + 9.91iT - 37T^{2} \) |
| 41 | \( 1 + 5.94iT - 41T^{2} \) |
| 43 | \( 1 + 1.13T + 43T^{2} \) |
| 47 | \( 1 - 6.81T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 4.23iT - 59T^{2} \) |
| 61 | \( 1 - 12.3iT - 61T^{2} \) |
| 67 | \( 1 + 5.74T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 8.27T + 73T^{2} \) |
| 79 | \( 1 - 4.86iT - 79T^{2} \) |
| 83 | \( 1 - 7.51iT - 83T^{2} \) |
| 89 | \( 1 - 4.44iT - 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.845802381948811701293380706059, −9.166526263387196373166107348576, −8.168726709287531322842467464829, −7.55719266080111627666613021052, −6.84349456617480654455393803364, −5.66569752466126536341503366958, −4.35904644183118548448010455427, −3.85378007537450817493837409191, −2.41365158890710828514210327233, −1.66903934415745459112757794998,
0.083886059601253307435160484804, 1.61577185905369433925472508608, 2.90443754043156097649368326883, 4.26207602404221054455772806598, 5.28923094549263383818768160817, 6.17770792256603258952133942011, 6.56214402800857511699360178940, 7.77360283312523259978619735855, 8.394106777577091160399791285748, 9.081259842361364111236993016785