| L(s) = 1 | − 5-s − 3.53i·11-s − 0.599i·13-s + 1.69·17-s − 7.78i·19-s + 3.31i·23-s + 25-s − 0.456i·29-s + 1.05i·31-s − 0.386·37-s + 0.478·41-s − 4.21·43-s − 8.55·47-s − 3.98i·53-s + 3.53i·55-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.06i·11-s − 0.166i·13-s + 0.411·17-s − 1.78i·19-s + 0.690i·23-s + 0.200·25-s − 0.0846i·29-s + 0.189i·31-s − 0.0634·37-s + 0.0746·41-s − 0.642·43-s − 1.24·47-s − 0.548i·53-s + 0.476i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7110843431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7110843431\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3.53iT - 11T^{2} \) |
| 13 | \( 1 + 0.599iT - 13T^{2} \) |
| 17 | \( 1 - 1.69T + 17T^{2} \) |
| 19 | \( 1 + 7.78iT - 19T^{2} \) |
| 23 | \( 1 - 3.31iT - 23T^{2} \) |
| 29 | \( 1 + 0.456iT - 29T^{2} \) |
| 31 | \( 1 - 1.05iT - 31T^{2} \) |
| 37 | \( 1 + 0.386T + 37T^{2} \) |
| 41 | \( 1 - 0.478T + 41T^{2} \) |
| 43 | \( 1 + 4.21T + 43T^{2} \) |
| 47 | \( 1 + 8.55T + 47T^{2} \) |
| 53 | \( 1 + 3.98iT - 53T^{2} \) |
| 59 | \( 1 - 2.60T + 59T^{2} \) |
| 61 | \( 1 + 11.0iT - 61T^{2} \) |
| 67 | \( 1 - 2.87T + 67T^{2} \) |
| 71 | \( 1 - 0.336iT - 71T^{2} \) |
| 73 | \( 1 + 3.08iT - 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 2.18iT - 97T^{2} \) |
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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
−7.50665249420963265057510705830, −6.72192578192706820810614738579, −6.17514341029445547005148316341, −5.19339871983887342607124377562, −4.82887377707771286922619054882, −3.67449403321880872530651957269, −3.24739301161819309704253570856, −2.34162909984228035699970546010, −1.11200854711268177718269834076, −0.17832029650219279202390169015,
1.28458539700699069135887779864, 2.09075877452166890297541818124, 3.10228266844231192840868845789, 3.91066623837938659854521325574, 4.48655518606255771088144927678, 5.28514936612525705973239919943, 6.04099355347630905425441760792, 6.77430785473304859623880535735, 7.42999397810628425393308718278, 8.035938627200035953161067866716
