L(s) = 1 | + 2.43i·2-s + 10.0·4-s − 25.9·5-s − 17.4·7-s + 63.4i·8-s − 63.0i·10-s + 44.7i·11-s − 157. i·13-s − 42.4i·14-s + 7.22·16-s + 328.·17-s + 678.·19-s − 261.·20-s − 108.·22-s − 424. i·23-s + ⋯ |
L(s) = 1 | + 0.607i·2-s + 0.630·4-s − 1.03·5-s − 0.356·7-s + 0.991i·8-s − 0.630i·10-s + 0.369i·11-s − 0.931i·13-s − 0.216i·14-s + 0.0282·16-s + 1.13·17-s + 1.87·19-s − 0.654·20-s − 0.224·22-s − 0.802i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.799361717\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.799361717\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 + (-1.10e3 - 3.30e3i)T \) |
good | 2 | \( 1 - 2.43iT - 16T^{2} \) |
| 5 | \( 1 + 25.9T + 625T^{2} \) |
| 7 | \( 1 + 17.4T + 2.40e3T^{2} \) |
| 11 | \( 1 - 44.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 157. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 328.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 678.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 424. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 870.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.38e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 2.56e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 503.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.00e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.42e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.52e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 5.70e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 4.77e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.79e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 4.71e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 8.66e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.45e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 4.00e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.91e3iT - 8.85e7T^{2} \) |
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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
−10.56918746942799110074638981339, −9.632130122253162215586696828433, −8.393125203470068366478146388351, −7.52451536170323763599880244690, −7.24689362411878290450922261230, −5.89446586091208148233316539673, −5.15261661312582494900010523529, −3.65588266327453976664582505088, −2.80100455632460139762245635005, −1.05867625242863830082145671116,
0.53198668927452106658593853367, 1.77714585177937276747740984393, 3.33829696757752954554760053544, 3.65950527359728770996784224439, 5.23606698955176069476715734990, 6.41080971707428111337242492993, 7.41336584251958442126061896585, 7.941131987991074362799828824724, 9.430262543551522619922688960547, 9.935492272956244851815475643132