| L(s) = 1 | + 2.82i·2-s − 5.19·3-s − 8.00·4-s − 27.9·5-s − 14.6i·6-s − 34.1·7-s − 22.6i·8-s + 27·9-s − 79.1i·10-s − 0.902i·11-s + 41.5·12-s − 282. i·13-s − 96.6i·14-s + 145.·15-s + 64.0·16-s − 311.·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.500·4-s − 1.11·5-s − 0.408i·6-s − 0.697·7-s − 0.353i·8-s + 0.333·9-s − 0.791i·10-s − 0.00746i·11-s + 0.288·12-s − 1.67i·13-s − 0.493i·14-s + 0.646·15-s + 0.250·16-s − 1.07·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4681077115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4681077115\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 + 5.19T \) |
| 59 | \( 1 + (3.45e3 - 453. i)T \) |
| good | 5 | \( 1 + 27.9T + 625T^{2} \) |
| 7 | \( 1 + 34.1T + 2.40e3T^{2} \) |
| 11 | \( 1 + 0.902iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 282. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 311.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 374.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 643. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.58e3T + 7.07e5T^{2} \) |
| 31 | \( 1 - 189. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 2.08e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.16e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 353. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.85e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.79e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 5.68e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 1.76e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 2.72e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 4.64e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 9.01e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.90e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.43e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.75e4iT - 8.85e7T^{2} \) |
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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
−11.07314660220402481686240493146, −10.21648931056247244801155441677, −9.111092347394318812685720246376, −8.010799155402384810595476749165, −7.33212642509664286718681474696, −6.30583557691560385145588383776, −5.32749569868915652709991068028, −4.20518141187775223020915177792, −3.09522703415340321587495936473, −0.62062100456471065368129545817,
0.26991320886287516938750753919, 1.91487111869747006040548841651, 3.60663121929842762062839245197, 4.23017228441836935600634176433, 5.54892859702128159491467065554, 6.86475726237013317414467244681, 7.63730215469309247510396785334, 9.121137109239219622399594638599, 9.569350827853100264037524269464, 11.01427756266734005886149526401
