| L(s) = 1 | + 0.736i·2-s + 3.45·4-s + 5.75i·5-s − 8.26·7-s + 5.49i·8-s − 4.23·10-s + 4.53i·11-s − 22.6·13-s − 6.08i·14-s + 9.77·16-s − 4.12i·17-s − 27.4·19-s + 19.8i·20-s − 3.33·22-s − 37.3i·23-s + ⋯ |
| L(s) = 1 | + 0.368i·2-s + 0.864·4-s + 1.15i·5-s − 1.18·7-s + 0.686i·8-s − 0.423·10-s + 0.411i·11-s − 1.74·13-s − 0.434i·14-s + 0.611·16-s − 0.242i·17-s − 1.44·19-s + 0.993i·20-s − 0.151·22-s − 1.62i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9164805512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9164805512\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + 4.12iT \) |
| good | 2 | \( 1 - 0.736iT - 4T^{2} \) |
| 5 | \( 1 - 5.75iT - 25T^{2} \) |
| 7 | \( 1 + 8.26T + 49T^{2} \) |
| 11 | \( 1 - 4.53iT - 121T^{2} \) |
| 13 | \( 1 + 22.6T + 169T^{2} \) |
| 19 | \( 1 + 27.4T + 361T^{2} \) |
| 23 | \( 1 + 37.3iT - 529T^{2} \) |
| 29 | \( 1 - 38.4iT - 841T^{2} \) |
| 31 | \( 1 - 34.1T + 961T^{2} \) |
| 37 | \( 1 + 0.534T + 1.36e3T^{2} \) |
| 41 | \( 1 + 9.07iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 18.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 54.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 69.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 35.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 4.75T + 3.72e3T^{2} \) |
| 67 | \( 1 - 53.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 84.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 109.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 112.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 162. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 44.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 96.2T + 9.40e3T^{2} \) |
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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
−11.09888768766211522696721548645, −10.37940204440678076259561585755, −9.796176353616508969884449274151, −8.437505981312936073541159701057, −7.13998846941102621390972981911, −6.85861338934766422015946323736, −6.07700460812135769809422036934, −4.61017050383210166773121253146, −2.94135745349559966183688394981, −2.43511763427268670274589798665,
0.33173497674309980247700466080, 2.00921978702794914095807950292, 3.19746855670605950671843763017, 4.46831538626933645386690922730, 5.72338789500307886460687235866, 6.62861103248316330190538947807, 7.61833838677988641699048301856, 8.695049732213388162021922166784, 9.798757236593912371996416363579, 10.13826457681079807105247680630
