| L(s) = 1 | + (−1.23 + 3.80i)2-s + 12.6·3-s + (−12.9 − 9.40i)4-s + (−20.2 − 14.7i)5-s + (−15.6 + 48.1i)6-s + (10.1 + 31.3i)7-s + (51.7 − 37.6i)8-s − 82.4·9-s + (81.0 − 58.8i)10-s + (−420. + 305. i)11-s + (−163. − 119. i)12-s + (−158. + 486. i)13-s − 132.·14-s + (−256. − 186. i)15-s + (79.1 + 243. i)16-s + (−664. + 482. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + 0.812·3-s + (−0.404 − 0.293i)4-s + (−0.362 − 0.263i)5-s + (−0.177 + 0.546i)6-s + (0.0786 + 0.242i)7-s + (0.286 − 0.207i)8-s − 0.339·9-s + (0.256 − 0.186i)10-s + (−1.04 + 0.761i)11-s + (−0.328 − 0.238i)12-s + (−0.259 + 0.799i)13-s − 0.180·14-s + (−0.294 − 0.213i)15-s + (0.0772 + 0.237i)16-s + (−0.557 + 0.405i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0184i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.00656195 + 0.711652i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00656195 + 0.711652i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 3.80i)T \) |
| 41 | \( 1 + (-2.49e3 + 1.04e4i)T \) |
| good | 3 | \( 1 - 12.6T + 243T^{2} \) |
| 5 | \( 1 + (20.2 + 14.7i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-10.1 - 31.3i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 11 | \( 1 + (420. - 305. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (158. - 486. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (664. - 482. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-406. - 1.25e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (5.36 - 16.5i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (2.43e3 + 1.76e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (2.90e3 - 2.11e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (392. + 284. i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 43 | \( 1 + (2.85e3 - 8.78e3i)T + (-1.18e8 - 8.64e7i)T^{2} \) |
| 47 | \( 1 + (-431. + 1.32e3i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-2.55e4 - 1.85e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (2.43e3 - 7.49e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-8.61e3 - 2.65e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-4.07e4 - 2.96e4i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (2.86e3 - 2.08e3i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + 3.94e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 632.T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.80e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + (1.54e4 + 4.74e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-8.73e4 - 6.34e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−14.12484346883950755473482703061, −13.04978079031348971885151798611, −11.84805933091264845886677690977, −10.29371513246769126210591712375, −9.095699256463873181874731848425, −8.218842282976331699022338499385, −7.26491137077239877745266771429, −5.61155111935808169562190229964, −4.14255243448306868831473854266, −2.23357057918075973925199399055,
0.27252378909298558620522654793, 2.52333249646521047736310452088, 3.49282075726298341012039743258, 5.31927265894945206713626565846, 7.43315857573687257950785472563, 8.336001324477777030949924560523, 9.406239933279739158449120408721, 10.74376884202994071863747573983, 11.46878692825515137386027766848, 13.02256518883375742414062560133
