| L(s) = 1 | + 2.85i·2-s − 4.14·4-s − 0.416i·8-s − 15.3·16-s − 19.8i·17-s − 37.8·19-s + 43.8i·23-s − 54.6·31-s − 45.6i·32-s + 56.6·34-s − 107. i·38-s − 125.·46-s − 14i·47-s − 49·49-s − 96.6i·53-s + ⋯ |
| L(s) = 1 | + 1.42i·2-s − 1.03·4-s − 0.0520i·8-s − 0.962·16-s − 1.16i·17-s − 1.99·19-s + 1.90i·23-s − 1.76·31-s − 1.42i·32-s + 1.66·34-s − 2.84i·38-s − 2.71·46-s − 0.297i·47-s − 0.999·49-s − 1.82i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2967234100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2967234100\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.85iT - 4T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 19.8iT - 289T^{2} \) |
| 19 | \( 1 + 37.8T + 361T^{2} \) |
| 23 | \( 1 - 43.8iT - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 54.6T + 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 + 14iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 96.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 85.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 + 58.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 23.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.00803502144157497625036146203, −9.755589899725455328741828537850, −8.975762257506236377585164626743, −8.164368438761147834679714682554, −7.30049483281221606648383595183, −6.65980557567309503677110705461, −5.65526676069406791891972985103, −4.93433558185813946411156925026, −3.71924777600301096274390545248, −2.09007486336941417049924461994,
0.099183599086412091551451960529, 1.69163884740504068733705538533, 2.59509884457021209879549047258, 3.86744268133190901136537797943, 4.52041726801165410068495175125, 6.03204734947177867195841543948, 6.89163535711664949822376311003, 8.311987362553912307681998278612, 8.917401932571953305990221141308, 9.974761085980506349394757658633
