| L(s) = 1 | − 3-s + (1.54 + 1.62i)5-s + i·7-s + 9-s + 5.83i·11-s + 3.72·13-s + (−1.54 − 1.62i)15-s − 6.90i·17-s − 2.96i·19-s − i·21-s + 7.94i·23-s + (−0.252 + 4.99i)25-s − 27-s + 4.46i·29-s + 7.61·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (0.689 + 0.724i)5-s + 0.377i·7-s + 0.333·9-s + 1.75i·11-s + 1.03·13-s + (−0.397 − 0.418i)15-s − 1.67i·17-s − 0.679i·19-s − 0.218i·21-s + 1.65i·23-s + (−0.0504 + 0.998i)25-s − 0.192·27-s + 0.829i·29-s + 1.36·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0318 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0318 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.804964426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804964426\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-1.54 - 1.62i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 5.83iT - 11T^{2} \) |
| 13 | \( 1 - 3.72T + 13T^{2} \) |
| 17 | \( 1 + 6.90iT - 17T^{2} \) |
| 19 | \( 1 + 2.96iT - 19T^{2} \) |
| 23 | \( 1 - 7.94iT - 23T^{2} \) |
| 29 | \( 1 - 4.46iT - 29T^{2} \) |
| 31 | \( 1 - 7.61T + 31T^{2} \) |
| 37 | \( 1 - 8.35T + 37T^{2} \) |
| 41 | \( 1 + 3.11T + 41T^{2} \) |
| 43 | \( 1 - 2.15T + 43T^{2} \) |
| 47 | \( 1 + 3.95iT - 47T^{2} \) |
| 53 | \( 1 - 7.52T + 53T^{2} \) |
| 59 | \( 1 + 4.90iT - 59T^{2} \) |
| 61 | \( 1 + 5.80iT - 61T^{2} \) |
| 67 | \( 1 - 1.13T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 - 6.68T + 83T^{2} \) |
| 89 | \( 1 + 9.72T + 89T^{2} \) |
| 97 | \( 1 - 9.50iT - 97T^{2} \) |
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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
−9.104985495624504089399974510384, −7.87364578433419339740136789120, −7.01002370776526210261688192715, −6.77740565199509726399354217845, −5.72312959139882720483680660622, −5.14400745445371941727075193049, −4.32285739234620752424027299626, −3.11927447988222905395402289518, −2.28501793331697985998056172986, −1.25138838827093563167535769338,
0.67675237935160856014342398127, 1.42101256462836483049005999782, 2.77139988604460605344018514790, 3.96919936256584967491523794404, 4.46316100135684550562830531153, 5.84330714943997257833197201927, 5.94313081147148737268199385771, 6.53411651143152782970399034679, 8.024128845055852815468421560055, 8.393534322303209543207353932576
