L(s) = 1 | − 1.61·2-s − 3-s + 0.618·4-s + 2.23·5-s + 1.61·6-s − 4.61·7-s + 2.23·8-s + 9-s − 3.61·10-s − 2.23·11-s − 0.618·12-s − 1.76·13-s + 7.47·14-s − 2.23·15-s − 4.85·16-s − 4.85·17-s − 1.61·18-s − 8.09·19-s + 1.38·20-s + 4.61·21-s + 3.61·22-s − 2.38·23-s − 2.23·24-s + 2.85·26-s − 27-s − 2.85·28-s + 8.61·29-s + ⋯ |
L(s) = 1 | − 1.14·2-s − 0.577·3-s + 0.309·4-s + 0.999·5-s + 0.660·6-s − 1.74·7-s + 0.790·8-s + 0.333·9-s − 1.14·10-s − 0.674·11-s − 0.178·12-s − 0.489·13-s + 1.99·14-s − 0.577·15-s − 1.21·16-s − 1.17·17-s − 0.381·18-s − 1.85·19-s + 0.309·20-s + 1.00·21-s + 0.771·22-s − 0.496·23-s − 0.456·24-s + 0.559·26-s − 0.192·27-s − 0.539·28-s + 1.60·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 + 8.09T + 19T^{2} \) |
| 23 | \( 1 + 2.38T + 23T^{2} \) |
| 29 | \( 1 - 8.61T + 29T^{2} \) |
| 31 | \( 1 - 9.56T + 31T^{2} \) |
| 37 | \( 1 + 6.85T + 37T^{2} \) |
| 41 | \( 1 + 3.09T + 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 47 | \( 1 + 4.14T + 47T^{2} \) |
| 53 | \( 1 - 1.76T + 53T^{2} \) |
| 61 | \( 1 + 9.85T + 61T^{2} \) |
| 67 | \( 1 + 2.70T + 67T^{2} \) |
| 71 | \( 1 - 9.94T + 71T^{2} \) |
| 73 | \( 1 + 5.85T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 - 0.618T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 3T + 97T^{2} \) |
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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
−12.25610894211952283070726136928, −10.52442417640008856256494621834, −10.22955949647475949358957612481, −9.371416471870876903956331230307, −8.390133296020820638503016036415, −6.78262173914783114446061021898, −6.20987270420816684246939244881, −4.57452023650093189296413504016, −2.38059346977599866708154952726, 0,
2.38059346977599866708154952726, 4.57452023650093189296413504016, 6.20987270420816684246939244881, 6.78262173914783114446061021898, 8.390133296020820638503016036415, 9.371416471870876903956331230307, 10.22955949647475949358957612481, 10.52442417640008856256494621834, 12.25610894211952283070726136928