| L(s) = 1 | − 2.22·5-s + 8.64·7-s − 19.8·11-s − 14.5i·13-s − 5.29i·17-s + 24.2i·19-s + 29.6i·23-s − 20.0·25-s − 6.13·29-s − 10.1·31-s − 19.2·35-s − 16.6i·37-s + 7.82i·41-s − 1.54i·43-s − 63.7i·47-s + ⋯ |
| L(s) = 1 | − 0.445·5-s + 1.23·7-s − 1.80·11-s − 1.11i·13-s − 0.311i·17-s + 1.27i·19-s + 1.28i·23-s − 0.801·25-s − 0.211·29-s − 0.326·31-s − 0.550·35-s − 0.449i·37-s + 0.190i·41-s − 0.0358i·43-s − 1.35i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1335660310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1335660310\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.22T + 25T^{2} \) |
| 7 | \( 1 - 8.64T + 49T^{2} \) |
| 11 | \( 1 + 19.8T + 121T^{2} \) |
| 13 | \( 1 + 14.5iT - 169T^{2} \) |
| 17 | \( 1 + 5.29iT - 289T^{2} \) |
| 19 | \( 1 - 24.2iT - 361T^{2} \) |
| 23 | \( 1 - 29.6iT - 529T^{2} \) |
| 29 | \( 1 + 6.13T + 841T^{2} \) |
| 31 | \( 1 + 10.1T + 961T^{2} \) |
| 37 | \( 1 + 16.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 7.82iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.54iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 63.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 76.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 41.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + 52.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 110. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 100. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 93.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 10.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 118.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 3.70T + 9.40e3T^{2} \) |
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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
−10.44683004026831033733588408520, −9.673194590574110850196674873812, −8.192841050494778276198736367857, −8.009072606145950569287940040619, −7.30714515209508691470337862183, −5.53969064460592012098529388906, −5.35573518919871071153381143759, −4.06213624527217085648599284309, −2.92306084403493514037447343184, −1.64784606085510650359731032357,
0.04156609398284835340856535092, 1.81773582886008482014253535255, 2.87172447686763588158539625953, 4.46733154737791901228258069817, 4.83352116750213091601468434062, 6.03484301983044502544605084038, 7.22115144618409607801292873262, 7.910783810289444039027322144071, 8.551485249488824377199457330669, 9.529911182224347994357300929527
