| L(s) = 1 | − 0.381·2-s − 2.61·3-s − 1.85·4-s − 3·5-s + 6-s + 2.85·7-s + 1.47·8-s + 3.85·9-s + 1.14·10-s − 2.23·11-s + 4.85·12-s − 6.85·13-s − 1.09·14-s + 7.85·15-s + 3.14·16-s − 5.23·17-s − 1.47·18-s + 3.85·19-s + 5.56·20-s − 7.47·21-s + 0.854·22-s − 1.47·23-s − 3.85·24-s + 4·25-s + 2.61·26-s − 2.23·27-s − 5.29·28-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 1.51·3-s − 0.927·4-s − 1.34·5-s + 0.408·6-s + 1.07·7-s + 0.520·8-s + 1.28·9-s + 0.362·10-s − 0.674·11-s + 1.40·12-s − 1.90·13-s − 0.291·14-s + 2.02·15-s + 0.786·16-s − 1.26·17-s − 0.346·18-s + 0.884·19-s + 1.24·20-s − 1.63·21-s + 0.182·22-s − 0.306·23-s − 0.786·24-s + 0.800·25-s + 0.513·26-s − 0.430·27-s − 1.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 67 | \( 1 + T \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 - 2.85T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 6.85T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 - 3.85T + 19T^{2} \) |
| 23 | \( 1 + 1.47T + 23T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 - 2.85T + 37T^{2} \) |
| 41 | \( 1 + 2.61T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 + 6.38T + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 6.56T + 61T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 0.326T + 83T^{2} \) |
| 89 | \( 1 - 2.23T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−14.42461883890019643490162312229, −12.81620168150096520907707440100, −11.83574552672041175960176229935, −11.14780217576772067169655962237, −9.927438659386322345236336345545, −8.203452589679141607491053854413, −7.26795772043444337656409462434, −5.11384122876646128717095438968, −4.54764937213672411163862912737, 0,
4.54764937213672411163862912737, 5.11384122876646128717095438968, 7.26795772043444337656409462434, 8.203452589679141607491053854413, 9.927438659386322345236336345545, 11.14780217576772067169655962237, 11.83574552672041175960176229935, 12.81620168150096520907707440100, 14.42461883890019643490162312229
