| L(s) = 1 | + (−2.68 + 1.33i)3-s + 5.84i·5-s + (5.43 − 7.17i)9-s + (−7.80 − 15.6i)15-s + 45.2i·23-s − 9.11·25-s + (−5 + 26.5i)27-s + 24.5·31-s − 72.8·37-s + (41.9 + 31.7i)45-s + 79.5i·47-s − 49·49-s + 79.5i·53-s − 67.7i·59-s + 129.·67-s + ⋯ |
| L(s) = 1 | + (−0.895 + 0.445i)3-s + 1.16i·5-s + (0.603 − 0.797i)9-s + (−0.520 − 1.04i)15-s + 1.96i·23-s − 0.364·25-s + (−0.185 + 0.982i)27-s + 0.793·31-s − 1.96·37-s + (0.931 + 0.704i)45-s + 1.69i·47-s − 0.999·49-s + 1.50i·53-s − 1.14i·59-s + 1.93·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5504953301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5504953301\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.68 - 1.33i)T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 5.84iT - 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 45.2iT - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 24.5T + 961T^{2} \) |
| 37 | \( 1 + 72.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 79.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 79.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 67.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 - 129.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 140. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 9.38iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 193.T + 9.40e3T^{2} \) |
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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
−9.822456216087685595216915018753, −9.348854068050792715866805532843, −8.040381118702175052374473158096, −7.14350350851665936071307765242, −6.54214963310822118265481541621, −5.71946534767333232536788335957, −4.88811464187788379400331844967, −3.75884748849938548751039450447, −3.01415605800957042766415882325, −1.49394680172105145221635480794,
0.19371232694311299921274228164, 1.16721091335841358240204923673, 2.32591060071653936732694682325, 3.96475007259221016816731187982, 4.91358716762315882347515097964, 5.35616643898398086809085157090, 6.48571913502075337741659707212, 7.03442830021469412352457621694, 8.371721937875354730460273302378, 8.519648367940843206853310676667
