| L(s) = 1 | + (2 + i)5-s − i·7-s − 4·11-s + 6i·13-s − 2i·17-s + 6·19-s − 2i·23-s + (3 + 4i)25-s + 6·29-s + 2·31-s + (1 − 2i)35-s − 4i·37-s − 8·41-s − 4i·43-s + 4i·47-s + ⋯ |
| L(s) = 1 | + (0.894 + 0.447i)5-s − 0.377i·7-s − 1.20·11-s + 1.66i·13-s − 0.485i·17-s + 1.37·19-s − 0.417i·23-s + (0.600 + 0.800i)25-s + 1.11·29-s + 0.359·31-s + (0.169 − 0.338i)35-s − 0.657i·37-s − 1.24·41-s − 0.609i·43-s + 0.583i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.089325744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.089325744\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 - i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.371422190880133544810976538365, −7.49108617024118605402482255855, −6.89827415027531138137850571021, −6.34017352639495177583777399026, −5.33140779341250005998173094597, −4.88621676502434601966023664023, −3.84408942843230488118187314515, −2.82907069222483250583880732454, −2.20657164050761643819472646699, −1.08012451450206714257082681843,
0.61118958819002827813211708971, 1.71178716529098566143770020686, 2.81433610684498022434565858824, 3.24851573616521442795489052846, 4.73086495946849708430634590333, 5.32163474782719265398877749467, 5.69278549745144952098938841180, 6.55970969453133521166372124521, 7.54027432318861647411878068266, 8.233049229312998812374115683297
