| L(s) = 1 | − 1.70·2-s − 2.43·3-s + 0.894·4-s − 4.04·5-s + 4.14·6-s + 4.09·7-s + 1.88·8-s + 2.94·9-s + 6.87·10-s − 11-s − 2.17·12-s + 13-s − 6.97·14-s + 9.85·15-s − 4.98·16-s + 3.40·17-s − 5.00·18-s − 0.396·19-s − 3.61·20-s − 9.99·21-s + 1.70·22-s + 1.73·23-s − 4.58·24-s + 11.3·25-s − 1.70·26-s + 0.142·27-s + 3.66·28-s + ⋯ |
| L(s) = 1 | − 1.20·2-s − 1.40·3-s + 0.447·4-s − 1.80·5-s + 1.69·6-s + 1.54·7-s + 0.665·8-s + 0.980·9-s + 2.17·10-s − 0.301·11-s − 0.629·12-s + 0.277·13-s − 1.86·14-s + 2.54·15-s − 1.24·16-s + 0.825·17-s − 1.17·18-s − 0.0909·19-s − 0.808·20-s − 2.18·21-s + 0.362·22-s + 0.360·23-s − 0.935·24-s + 2.26·25-s − 0.333·26-s + 0.0274·27-s + 0.692·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2969561144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2969561144\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 3 | \( 1 + 2.43T + 3T^{2} \) |
| 5 | \( 1 + 4.04T + 5T^{2} \) |
| 7 | \( 1 - 4.09T + 7T^{2} \) |
| 17 | \( 1 - 3.40T + 17T^{2} \) |
| 19 | \( 1 + 0.396T + 19T^{2} \) |
| 23 | \( 1 - 1.73T + 23T^{2} \) |
| 29 | \( 1 - 7.08T + 29T^{2} \) |
| 31 | \( 1 + 8.11T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 - 1.88T + 41T^{2} \) |
| 43 | \( 1 - 2.07T + 43T^{2} \) |
| 47 | \( 1 - 3.47T + 47T^{2} \) |
| 53 | \( 1 - 6.80T + 53T^{2} \) |
| 59 | \( 1 - 3.45T + 59T^{2} \) |
| 61 | \( 1 - 4.40T + 61T^{2} \) |
| 67 | \( 1 + 1.82T + 67T^{2} \) |
| 71 | \( 1 + 2.68T + 71T^{2} \) |
| 73 | \( 1 - 9.07T + 73T^{2} \) |
| 79 | \( 1 - 9.08T + 79T^{2} \) |
| 83 | \( 1 + 9.37T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 5.24T + 97T^{2} \) |
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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
−12.53273335972217911220094415131, −11.61060956612669847274399967728, −11.11727155247172169211834552983, −10.44350262431217324909929058376, −8.661896152930553256698689829024, −7.905970450383018768008998742195, −7.14069032647997674078984716140, −5.22316239607282973845728353366, −4.27417280418363403604014638377, −0.876912812617871231683189995997,
0.876912812617871231683189995997, 4.27417280418363403604014638377, 5.22316239607282973845728353366, 7.14069032647997674078984716140, 7.905970450383018768008998742195, 8.661896152930553256698689829024, 10.44350262431217324909929058376, 11.11727155247172169211834552983, 11.61060956612669847274399967728, 12.53273335972217911220094415131
