| L(s) = 1 | + 6.55e4·2-s − 7.12e7·3-s + 4.29e9·4-s + 1.52e11·5-s − 4.66e12·6-s − 7.62e13·7-s + 2.81e14·8-s − 4.88e14·9-s + 1.00e16·10-s + 1.14e17·11-s − 3.05e17·12-s + 3.91e18·13-s − 4.99e18·14-s − 1.08e19·15-s + 1.84e19·16-s − 1.94e20·17-s − 3.20e19·18-s − 1.21e21·19-s + 6.55e20·20-s + 5.43e21·21-s + 7.49e21·22-s − 4.03e21·23-s − 2.00e22·24-s + 2.32e22·25-s + 2.56e23·26-s + 4.30e23·27-s − 3.27e23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.955·3-s + 0.5·4-s + 0.447·5-s − 0.675·6-s − 0.867·7-s + 0.353·8-s − 0.0878·9-s + 0.316·10-s + 0.750·11-s − 0.477·12-s + 1.63·13-s − 0.613·14-s − 0.427·15-s + 0.250·16-s − 0.970·17-s − 0.0621·18-s − 0.969·19-s + 0.223·20-s + 0.828·21-s + 0.530·22-s − 0.137·23-s − 0.337·24-s + 0.200·25-s + 1.15·26-s + 1.03·27-s − 0.433·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 6.55e4T \) |
| 5 | \( 1 - 1.52e11T \) |
| good | 3 | \( 1 + 7.12e7T + 5.55e15T^{2} \) |
| 7 | \( 1 + 7.62e13T + 7.73e27T^{2} \) |
| 11 | \( 1 - 1.14e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 3.91e18T + 5.75e36T^{2} \) |
| 17 | \( 1 + 1.94e20T + 4.02e40T^{2} \) |
| 19 | \( 1 + 1.21e21T + 1.58e42T^{2} \) |
| 23 | \( 1 + 4.03e21T + 8.65e44T^{2} \) |
| 29 | \( 1 - 1.32e23T + 1.81e48T^{2} \) |
| 31 | \( 1 - 4.55e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 4.62e25T + 5.63e51T^{2} \) |
| 41 | \( 1 + 4.25e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 5.95e26T + 8.02e53T^{2} \) |
| 47 | \( 1 + 3.08e27T + 1.51e55T^{2} \) |
| 53 | \( 1 + 5.55e28T + 7.96e56T^{2} \) |
| 59 | \( 1 + 1.85e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 1.55e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 2.16e30T + 1.82e60T^{2} \) |
| 71 | \( 1 + 3.10e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 7.37e30T + 3.08e61T^{2} \) |
| 79 | \( 1 - 6.89e30T + 4.18e62T^{2} \) |
| 83 | \( 1 - 5.09e31T + 2.13e63T^{2} \) |
| 89 | \( 1 + 2.16e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 1.46e32T + 3.65e65T^{2} \) |
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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
−12.85118011368822357487856761933, −11.55596914031915609647500578712, −10.52376932753410555792810757421, −8.800970544287820777189424477440, −6.35489110182631321256905923617, −6.23422965452623027649869639468, −4.54955443710652652625236884855, −3.18021190696223331447625657936, −1.49525265437449679102209993794, 0,
1.49525265437449679102209993794, 3.18021190696223331447625657936, 4.54955443710652652625236884855, 6.23422965452623027649869639468, 6.35489110182631321256905923617, 8.800970544287820777189424477440, 10.52376932753410555792810757421, 11.55596914031915609647500578712, 12.85118011368822357487856761933
