| L(s) = 1 | − 3.45·3-s − 3.43·5-s − 4.02·7-s + 8.92·9-s − 0.264·11-s − 6.04·13-s + 11.8·15-s − 0.684·17-s − 2.21·19-s + 13.9·21-s + 1.35·23-s + 6.80·25-s − 20.4·27-s − 10.5·29-s − 6.55·31-s + 0.913·33-s + 13.8·35-s − 7.81·37-s + 20.8·39-s + 3.36·41-s − 0.439·43-s − 30.6·45-s + 2.99·47-s + 9.23·49-s + 2.36·51-s − 0.507·53-s + 0.908·55-s + ⋯ |
| L(s) = 1 | − 1.99·3-s − 1.53·5-s − 1.52·7-s + 2.97·9-s − 0.0797·11-s − 1.67·13-s + 3.06·15-s − 0.165·17-s − 0.507·19-s + 3.03·21-s + 0.281·23-s + 1.36·25-s − 3.93·27-s − 1.95·29-s − 1.17·31-s + 0.158·33-s + 2.34·35-s − 1.28·37-s + 3.34·39-s + 0.525·41-s − 0.0669·43-s − 4.57·45-s + 0.436·47-s + 1.31·49-s + 0.330·51-s − 0.0696·53-s + 0.122·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 547 | \( 1 - T \) |
| good | 3 | \( 1 + 3.45T + 3T^{2} \) |
| 5 | \( 1 + 3.43T + 5T^{2} \) |
| 7 | \( 1 + 4.02T + 7T^{2} \) |
| 11 | \( 1 + 0.264T + 11T^{2} \) |
| 13 | \( 1 + 6.04T + 13T^{2} \) |
| 17 | \( 1 + 0.684T + 17T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 6.55T + 31T^{2} \) |
| 37 | \( 1 + 7.81T + 37T^{2} \) |
| 41 | \( 1 - 3.36T + 41T^{2} \) |
| 43 | \( 1 + 0.439T + 43T^{2} \) |
| 47 | \( 1 - 2.99T + 47T^{2} \) |
| 53 | \( 1 + 0.507T + 53T^{2} \) |
| 59 | \( 1 - 6.16T + 59T^{2} \) |
| 61 | \( 1 - 6.91T + 61T^{2} \) |
| 67 | \( 1 - 4.63T + 67T^{2} \) |
| 71 | \( 1 - 0.550T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 1.58T + 79T^{2} \) |
| 83 | \( 1 + 6.71T + 83T^{2} \) |
| 89 | \( 1 + 5.96T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.21039268128259550000508083958, −6.90599052405431288508908840356, −5.99590735897357459058544064513, −5.38932232847911098408315866912, −4.69299562055727743083510746521, −3.97020551230408549955900678228, −3.43480686313426004482514408976, −2.06678938982420419790901529519, −0.49902458333977928369763541796, 0,
0.49902458333977928369763541796, 2.06678938982420419790901529519, 3.43480686313426004482514408976, 3.97020551230408549955900678228, 4.69299562055727743083510746521, 5.38932232847911098408315866912, 5.99590735897357459058544064513, 6.90599052405431288508908840356, 7.21039268128259550000508083958
