| L(s) = 1 | + 2.41·3-s + (−1.94 − 1.10i)5-s + i·7-s + 2.84·9-s − 2.55i·11-s − 5.47·13-s + (−4.70 − 2.66i)15-s − 2.44i·17-s − 4.99i·19-s + 2.41i·21-s − 1.10i·23-s + (2.56 + 4.29i)25-s − 0.376·27-s + 1.96i·29-s + 2.33·31-s + ⋯ |
| L(s) = 1 | + 1.39·3-s + (−0.869 − 0.493i)5-s + 0.377i·7-s + 0.948·9-s − 0.770i·11-s − 1.51·13-s + (−1.21 − 0.688i)15-s − 0.593i·17-s − 1.14i·19-s + 0.527i·21-s − 0.229i·23-s + (0.513 + 0.858i)25-s − 0.0724·27-s + 0.364i·29-s + 0.419·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.222553932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.222553932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.94 + 1.10i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 11 | \( 1 + 2.55iT - 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 + 2.44iT - 17T^{2} \) |
| 19 | \( 1 + 4.99iT - 19T^{2} \) |
| 23 | \( 1 + 1.10iT - 23T^{2} \) |
| 29 | \( 1 - 1.96iT - 29T^{2} \) |
| 31 | \( 1 - 2.33T + 31T^{2} \) |
| 37 | \( 1 + 6.16T + 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 + 5.99T + 43T^{2} \) |
| 47 | \( 1 + 8.05iT - 47T^{2} \) |
| 53 | \( 1 - 1.97T + 53T^{2} \) |
| 59 | \( 1 - 1.44iT - 59T^{2} \) |
| 61 | \( 1 + 8.71iT - 61T^{2} \) |
| 67 | \( 1 - 3.92T + 67T^{2} \) |
| 71 | \( 1 - 4.55T + 71T^{2} \) |
| 73 | \( 1 + 7.55iT - 73T^{2} \) |
| 79 | \( 1 + 8.53T + 79T^{2} \) |
| 83 | \( 1 - 8.26T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 10.6iT - 97T^{2} \) |
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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
−8.617352433347179111018152541443, −8.229058369057825064487662858673, −7.34248937265265754576586808191, −6.78778446954408072371039435520, −5.22907645322417031472508854330, −4.74299391103863308195857416737, −3.54490581147559571871033735400, −2.97094123421887096273286253635, −2.03351152924021335692200620848, −0.32068132532766319921829394194,
1.77743567010856875093919372652, 2.68933709917837897272215058283, 3.55181690309822639209773686548, 4.19218768537373883699834243246, 5.13113264181861159254585230573, 6.56073497017471801573790251122, 7.28297617891842050877028331892, 7.84821962481211012649040090574, 8.328445024307145788885420471093, 9.285246643917882651434898859296
