| L(s) = 1 | + 18.8·5-s − 33.6i·7-s + 43.3i·11-s + 140.·13-s + 90.3·17-s − 131. i·19-s − 771. i·23-s − 270.·25-s + 523.·29-s + 754. i·31-s − 634. i·35-s − 743.·37-s + 2.40e3·41-s − 347. i·43-s − 2.12e3i·47-s + ⋯ |
| L(s) = 1 | + 0.753·5-s − 0.687i·7-s + 0.358i·11-s + 0.832·13-s + 0.312·17-s − 0.363i·19-s − 1.45i·23-s − 0.432·25-s + 0.622·29-s + 0.784i·31-s − 0.517i·35-s − 0.542·37-s + 1.43·41-s − 0.187i·43-s − 0.962i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.289726988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.289726988\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 18.8T + 625T^{2} \) |
| 7 | \( 1 + 33.6iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 43.3iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 140.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 90.3T + 8.35e4T^{2} \) |
| 19 | \( 1 + 131. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 771. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 523.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 754. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 743.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.40e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 347. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.12e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.66e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 6.59e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 749.T + 1.38e7T^{2} \) |
| 67 | \( 1 + 7.83e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 3.05e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 320.T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.05e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 7.22e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 9.67e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.00e4T + 8.85e7T^{2} \) |
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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
−10.79135965587595664070837376975, −10.24866646162398854498455253772, −9.200999631987918602529969583326, −8.222772081126026334302491484496, −7.00796997288028794946707956940, −6.15223812073666247854511161374, −4.93949868232261897672738727192, −3.73201346524809136931817352196, −2.22602259003357516068134857289, −0.811983526796628426162440536347,
1.26348854679491824277439890187, 2.59269387163415603676605122657, 3.95077045362829066328973052715, 5.60863593544882314276556436344, 5.97881601399155091228154349704, 7.42969668477659085000243508977, 8.556837848699667283718403071778, 9.377306064065981473642055673350, 10.26488343757697180341403449713, 11.33485119603122762558274974442
