| L(s) = 1 | − 7-s + 4·11-s − 6·17-s + 6·19-s − 23-s − 5·25-s − 10·29-s − 4·31-s − 2·37-s + 10·41-s + 4·43-s + 12·47-s + 49-s + 6·53-s − 2·59-s − 8·71-s − 6·73-s − 4·77-s + 8·79-s − 14·83-s + 14·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.20·11-s − 1.45·17-s + 1.37·19-s − 0.208·23-s − 25-s − 1.85·29-s − 0.718·31-s − 0.328·37-s + 1.56·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 0.260·59-s − 0.949·71-s − 0.702·73-s − 0.455·77-s + 0.900·79-s − 1.53·83-s + 1.48·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.72950470218635, −15.24759500779522, −14.66803710075810, −13.99769833174984, −13.71514810632436, −13.00327419398758, −12.57329374624118, −11.75666041226326, −11.51162136131638, −10.91698686271500, −10.23738042525228, −9.498560363883029, −9.060324125865292, −8.885752971174494, −7.643224468042108, −7.420129740066058, −6.769661010114685, −5.903630939737395, −5.742361420451613, −4.726372048067571, −3.848560262536917, −3.789907549435840, −2.628924016777895, −1.960588926254689, −1.077645851967392, 0,
1.077645851967392, 1.960588926254689, 2.628924016777895, 3.789907549435840, 3.848560262536917, 4.726372048067571, 5.742361420451613, 5.903630939737395, 6.769661010114685, 7.420129740066058, 7.643224468042108, 8.885752971174494, 9.060324125865292, 9.498560363883029, 10.23738042525228, 10.91698686271500, 11.51162136131638, 11.75666041226326, 12.57329374624118, 13.00327419398758, 13.71514810632436, 13.99769833174984, 14.66803710075810, 15.24759500779522, 15.72950470218635
