L(s) = 1 | + 0.133i·2-s + 14.8i·3-s + 15.9·4-s − 26.0i·5-s − 1.97·6-s + 87.9i·7-s + 4.26i·8-s − 138.·9-s + 3.47·10-s − 101.·11-s + 236. i·12-s + 190.·13-s − 11.7·14-s + 385.·15-s + 255.·16-s − 227.·17-s + ⋯ |
L(s) = 1 | + 0.0333i·2-s + 1.64i·3-s + 0.998·4-s − 1.04i·5-s − 0.0548·6-s + 1.79i·7-s + 0.0666i·8-s − 1.70·9-s + 0.0347·10-s − 0.840·11-s + 1.64i·12-s + 1.12·13-s − 0.0598·14-s + 1.71·15-s + 0.996·16-s − 0.788·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0745 - 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0745 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.17485 + 1.26590i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17485 + 1.26590i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (137. + 1.84e3i)T \) |
good | 2 | \( 1 - 0.133iT - 16T^{2} \) |
| 3 | \( 1 - 14.8iT - 81T^{2} \) |
| 5 | \( 1 + 26.0iT - 625T^{2} \) |
| 7 | \( 1 - 87.9iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 101.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 190.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 227.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 304. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 797.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 1.08e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 433.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 310. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.33e3T + 2.82e6T^{2} \) |
| 47 | \( 1 - 1.00e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 3.64e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.28e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 2.27e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 4.75e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.25e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.32e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 227.T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.89e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.08e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.09e4T + 8.85e7T^{2} \) |
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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
−15.60118151551328094046641428777, −15.14143718400814314375438755455, −13.00695844548918930409503600616, −11.65890310106071817257822446312, −10.78457474043951332442525377739, −9.223846148068503352117825418777, −8.531536419778971310727590575771, −5.89833870847558574109162555497, −4.86267442932936355387105180397, −2.76516038077872148322488566760,
1.27544788562586107829633175294, 3.06927732898238882247780417775, 6.40067771180259219568478316355, 7.07465846830812508929073911249, 7.891037933038580423995626443439, 10.74471567353136241406644807939, 11.02484178992664422982967872940, 12.74312065542155527304198033460, 13.59329147859155880946878761708, 14.60420708558352966523831277305