| L(s) = 1 | + 1.53·2-s + 1.34·3-s + 0.347·4-s + 3.53·5-s + 2.06·6-s − 0.347·7-s − 2.53·8-s − 1.18·9-s + 5.41·10-s − 1.75·11-s + 0.467·12-s − 3.29·13-s − 0.532·14-s + 4.75·15-s − 4.57·16-s − 1.81·18-s + 1.53·19-s + 1.22·20-s − 0.467·21-s − 2.69·22-s + 2.81·23-s − 3.41·24-s + 7.47·25-s − 5.04·26-s − 5.63·27-s − 0.120·28-s − 1.18·29-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.777·3-s + 0.173·4-s + 1.57·5-s + 0.842·6-s − 0.131·7-s − 0.895·8-s − 0.394·9-s + 1.71·10-s − 0.530·11-s + 0.135·12-s − 0.912·13-s − 0.142·14-s + 1.22·15-s − 1.14·16-s − 0.427·18-s + 0.351·19-s + 0.274·20-s − 0.102·21-s − 0.574·22-s + 0.587·23-s − 0.696·24-s + 1.49·25-s − 0.988·26-s − 1.08·27-s − 0.0227·28-s − 0.220·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.715784905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.715784905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 3 | \( 1 - 1.34T + 3T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 7 | \( 1 + 0.347T + 7T^{2} \) |
| 11 | \( 1 + 1.75T + 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 23 | \( 1 - 2.81T + 23T^{2} \) |
| 29 | \( 1 + 1.18T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 + 3.92T + 37T^{2} \) |
| 41 | \( 1 - 4.92T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 5.12T + 47T^{2} \) |
| 53 | \( 1 + 8.36T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 4.41T + 61T^{2} \) |
| 67 | \( 1 - 8.07T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 1.70T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 2.14T + 83T^{2} \) |
| 89 | \( 1 - 6.41T + 89T^{2} \) |
| 97 | \( 1 - 4.08T + 97T^{2} \) |
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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
−12.25062355585239086890345454046, −10.89324247861565494697078190826, −9.598095766565214398668241800015, −9.280991654328322591458734917650, −7.997572557449682599120900217150, −6.51550295346740496705565060157, −5.59970911349892280977732611846, −4.79330285029268887785925022116, −3.13062909427611800244818372476, −2.35117265347252672011920194772,
2.35117265347252672011920194772, 3.13062909427611800244818372476, 4.79330285029268887785925022116, 5.59970911349892280977732611846, 6.51550295346740496705565060157, 7.997572557449682599120900217150, 9.280991654328322591458734917650, 9.598095766565214398668241800015, 10.89324247861565494697078190826, 12.25062355585239086890345454046
