| L(s) = 1 | + (0.5 − 1.93i)2-s + (−3.50 − 1.93i)4-s + (−5.50 + 5.80i)8-s + (8.50 + 13.5i)16-s + 14·17-s + 30.9i·19-s + 30.9i·23-s − 61.9i·31-s + (30.5 − 9.68i)32-s + (7 − 27.1i)34-s + (60.0 + 15.4i)38-s + (60.0 + 15.4i)46-s + 92.9i·47-s + 49·49-s + 86·53-s + ⋯ |
| L(s) = 1 | + (0.250 − 0.968i)2-s + (−0.875 − 0.484i)4-s + (−0.687 + 0.726i)8-s + (0.531 + 0.847i)16-s + 0.823·17-s + 1.63i·19-s + 1.34i·23-s − 1.99i·31-s + (0.953 − 0.302i)32-s + (0.205 − 0.797i)34-s + (1.57 + 0.407i)38-s + (1.30 + 0.336i)46-s + 1.97i·47-s + 0.999·49-s + 1.62·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.484i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.875 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.748149278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.748149278\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.5 + 1.93i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 14T + 289T^{2} \) |
| 19 | \( 1 - 30.9iT - 361T^{2} \) |
| 23 | \( 1 - 30.9iT - 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 + 61.9iT - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 92.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 86T + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 118T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 + 123. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 61.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3T^{2} \) |
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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
−9.891384690975723758663312508526, −9.364562297957143705828456821191, −8.227685420935828999900138046427, −7.52689278994781506594991910252, −5.97865886918113664478881843963, −5.48103353130138898002890199466, −4.17319792296347694424780866894, −3.47715187727858018604461441177, −2.22972943008531047637106077038, −1.05562944383288945052537382242,
0.65790968067746589783569440519, 2.68617202105089289588207724228, 3.82749204806894224196570705590, 4.88701795314231003894735663404, 5.55168824246522250293868340000, 6.80250495111581684565540076251, 7.10490609286310707001461012683, 8.426564959904299688452739566144, 8.769321032484791338039519622630, 9.864473179883106180022424344895
