| L(s) = 1 | − 1.55i·5-s + 4.19·7-s + 15.8i·11-s + 9.39·13-s − 23.9i·17-s + 18.9·19-s − 25.1i·23-s + 22.5·25-s + 55.2i·29-s + 39.1·31-s − 6.51i·35-s + 19.1·37-s − 2.80i·41-s + 33.7·43-s + 16.1i·47-s + ⋯ |
| L(s) = 1 | − 0.310i·5-s + 0.599·7-s + 1.43i·11-s + 0.722·13-s − 1.40i·17-s + 0.998·19-s − 1.09i·23-s + 0.903·25-s + 1.90i·29-s + 1.26·31-s − 0.186i·35-s + 0.518·37-s − 0.0683i·41-s + 0.785·43-s + 0.343i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.84243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84243\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.55iT - 25T^{2} \) |
| 7 | \( 1 - 4.19T + 49T^{2} \) |
| 11 | \( 1 - 15.8iT - 121T^{2} \) |
| 13 | \( 1 - 9.39T + 169T^{2} \) |
| 17 | \( 1 + 23.9iT - 289T^{2} \) |
| 19 | \( 1 - 18.9T + 361T^{2} \) |
| 23 | \( 1 + 25.1iT - 529T^{2} \) |
| 29 | \( 1 - 55.2iT - 841T^{2} \) |
| 31 | \( 1 - 39.1T + 961T^{2} \) |
| 37 | \( 1 - 19.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 2.80iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 33.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 16.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 53.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 9.92iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 62.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 20.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 113. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 110.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 40.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + 164. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 13.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 154.T + 9.40e3T^{2} \) |
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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
−11.46464818644992980246921158374, −10.46843976449011184613352618164, −9.488867508759330702300204173544, −8.631304684732611329347005333085, −7.52265305052843310296053383672, −6.69662764357629439927133519543, −5.15433083830484049087304273826, −4.51991012831421237455078005825, −2.83397675888585095071214796323, −1.23599859743626307246974815352,
1.19735404721171998666768906021, 2.99843533942814589018811870091, 4.13617903638453728267983906186, 5.63461406174586488362283557152, 6.33386686322493924054873284539, 7.81682792424688675820162605131, 8.393912694211232332621122928540, 9.521846276919860956664870286524, 10.71360166896402505764411410837, 11.24949361617741493184968554661
