| L(s) = 1 | − 2·4-s + 4.25i·5-s + 3.31i·11-s + 4·16-s − 8.51i·20-s − 3.31i·23-s − 13.1·25-s − 11.1·31-s − 5.11·37-s − 6.63i·44-s + 7.07i·47-s + 7·49-s + 1.43i·53-s − 14.1·55-s − 11.3i·59-s + ⋯ |
| L(s) = 1 | − 4-s + 1.90i·5-s + 1.00i·11-s + 16-s − 1.90i·20-s − 0.691i·23-s − 2.62·25-s − 1.99·31-s − 0.841·37-s − 1.00i·44-s + 1.03i·47-s + 49-s + 0.197i·53-s − 1.90·55-s − 1.47i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.638917i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.638917i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - 3.31iT \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 - 4.25iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 3.31iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 11.1T + 31T^{2} \) |
| 37 | \( 1 + 5.11T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 7.07iT - 47T^{2} \) |
| 53 | \( 1 - 1.43iT - 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 2.11T + 67T^{2} \) |
| 71 | \( 1 + 5.69iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 16.5iT - 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−10.49490204589582780143755800383, −9.799206561339967031658437295465, −9.040766024813352618538424115417, −7.82727229240121470074088526289, −7.19121234182572331601111758609, −6.34631513051654502852748216419, −5.29486193471198034119719035560, −4.12243622344941705737656843143, −3.30036534136469064049827036123, −2.10754483518850064835032181194,
0.32353526567912765520234644573, 1.51092198647333421030637917437, 3.57175837889343953216801863367, 4.34429847221281615497940346562, 5.41755835543497959201848098790, 5.64858788963120956434256139082, 7.41637424366764641038880190355, 8.385621403547732644708244291160, 8.848612661810696867028238062451, 9.375766504620632782841784241517
