L(s) = 1 | − 1.61·2-s + 2.23·3-s + 0.618·4-s − 3.23·5-s − 3.61·6-s − 1.23·7-s + 2.23·8-s + 2.00·9-s + 5.23·10-s − 0.763·11-s + 1.38·12-s + 3·13-s + 2.00·14-s − 7.23·15-s − 4.85·16-s + 5.23·17-s − 3.23·18-s − 2·19-s − 2.00·20-s − 2.76·21-s + 1.23·22-s + 23-s + 5.00·24-s + 5.47·25-s − 4.85·26-s − 2.23·27-s − 0.763·28-s + ⋯ |
L(s) = 1 | − 1.14·2-s + 1.29·3-s + 0.309·4-s − 1.44·5-s − 1.47·6-s − 0.467·7-s + 0.790·8-s + 0.666·9-s + 1.65·10-s − 0.230·11-s + 0.398·12-s + 0.832·13-s + 0.534·14-s − 1.86·15-s − 1.21·16-s + 1.26·17-s − 0.762·18-s − 0.458·19-s − 0.447·20-s − 0.603·21-s + 0.263·22-s + 0.208·23-s + 1.02·24-s + 1.09·25-s − 0.951·26-s − 0.430·27-s − 0.144·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4503793707\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4503793707\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 - T \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 - 3.23T + 37T^{2} \) |
| 41 | \( 1 - 5.47T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 2.47T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 7.23T + 67T^{2} \) |
| 71 | \( 1 - 7.76T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 - 6.94T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 - 4.29T + 97T^{2} \) |
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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
−18.47750850800061646009902647786, −16.63291171064781582535205855744, −15.65538270528692681049757116794, −14.40066184005226474772481053609, −12.90390695537994036129091050107, −11.06445498538988679371642241803, −9.462792302698072156583043638081, −8.321948106011545948491366974293, −7.56305052056930034956656701516, −3.67521748643984848866430372052,
3.67521748643984848866430372052, 7.56305052056930034956656701516, 8.321948106011545948491366974293, 9.462792302698072156583043638081, 11.06445498538988679371642241803, 12.90390695537994036129091050107, 14.40066184005226474772481053609, 15.65538270528692681049757116794, 16.63291171064781582535205855744, 18.47750850800061646009902647786