| L(s) = 1 | − 1.70·3-s + 3.66·5-s + 7-s − 0.0802·9-s + 1.52·11-s − 4.21·13-s − 6.26·15-s − 7.18·17-s − 3.19·19-s − 1.70·21-s + 3.49·23-s + 8.45·25-s + 5.26·27-s + 1.20·29-s + 3.95·31-s − 2.61·33-s + 3.66·35-s − 8.38·37-s + 7.19·39-s + 2.67·41-s + 6.01·43-s − 0.294·45-s + 2.35·47-s + 49-s + 12.2·51-s − 4.83·53-s + 5.60·55-s + ⋯ |
| L(s) = 1 | − 0.986·3-s + 1.64·5-s + 0.377·7-s − 0.0267·9-s + 0.460·11-s − 1.16·13-s − 1.61·15-s − 1.74·17-s − 0.732·19-s − 0.372·21-s + 0.729·23-s + 1.69·25-s + 1.01·27-s + 0.223·29-s + 0.710·31-s − 0.454·33-s + 0.620·35-s − 1.37·37-s + 1.15·39-s + 0.417·41-s + 0.917·43-s − 0.0439·45-s + 0.343·47-s + 0.142·49-s + 1.71·51-s − 0.663·53-s + 0.755·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 5 | \( 1 - 3.66T + 5T^{2} \) |
| 11 | \( 1 - 1.52T + 11T^{2} \) |
| 13 | \( 1 + 4.21T + 13T^{2} \) |
| 17 | \( 1 + 7.18T + 17T^{2} \) |
| 19 | \( 1 + 3.19T + 19T^{2} \) |
| 23 | \( 1 - 3.49T + 23T^{2} \) |
| 29 | \( 1 - 1.20T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 37 | \( 1 + 8.38T + 37T^{2} \) |
| 41 | \( 1 - 2.67T + 41T^{2} \) |
| 43 | \( 1 - 6.01T + 43T^{2} \) |
| 47 | \( 1 - 2.35T + 47T^{2} \) |
| 53 | \( 1 + 4.83T + 53T^{2} \) |
| 59 | \( 1 - 5.46T + 59T^{2} \) |
| 61 | \( 1 - 1.98T + 61T^{2} \) |
| 67 | \( 1 + 8.06T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 3.32T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 1.70T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 1.08T + 97T^{2} \) |
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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
−7.20327312201593458675745256067, −6.75074892218278893208934061982, −6.09351996145136952264712803790, −5.58463571182856550391044609949, −4.79862336476373462529156285724, −4.40988702922870900213190017291, −2.80212961564441690164004595245, −2.23008932478638475160697870623, −1.32396183443581773759774440585, 0,
1.32396183443581773759774440585, 2.23008932478638475160697870623, 2.80212961564441690164004595245, 4.40988702922870900213190017291, 4.79862336476373462529156285724, 5.58463571182856550391044609949, 6.09351996145136952264712803790, 6.75074892218278893208934061982, 7.20327312201593458675745256067
