| L(s) = 1 | − 5.07·2-s + 1.55·3-s + 17.7·4-s + 10.0·5-s − 7.91·6-s + 24.3·7-s − 49.6·8-s − 24.5·9-s − 51.0·10-s + 1.55·11-s + 27.7·12-s + 85.6·13-s − 123.·14-s + 15.6·15-s + 109.·16-s − 35.1·17-s + 124.·18-s − 124.·19-s + 178.·20-s + 37.9·21-s − 7.91·22-s − 23·23-s − 77.4·24-s − 23.7·25-s − 434.·26-s − 80.3·27-s + 432.·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.299·3-s + 2.22·4-s + 0.900·5-s − 0.538·6-s + 1.31·7-s − 2.19·8-s − 0.910·9-s − 1.61·10-s + 0.0427·11-s + 0.666·12-s + 1.82·13-s − 2.35·14-s + 0.270·15-s + 1.71·16-s − 0.500·17-s + 1.63·18-s − 1.50·19-s + 2.00·20-s + 0.394·21-s − 0.0766·22-s − 0.208·23-s − 0.658·24-s − 0.189·25-s − 3.27·26-s − 0.572·27-s + 2.92·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6921848951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6921848951\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 5.07T + 8T^{2} \) |
| 3 | \( 1 - 1.55T + 27T^{2} \) |
| 5 | \( 1 - 10.0T + 125T^{2} \) |
| 7 | \( 1 - 24.3T + 343T^{2} \) |
| 11 | \( 1 - 1.55T + 1.33e3T^{2} \) |
| 13 | \( 1 - 85.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 35.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 124.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 130.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 82.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 107.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 35.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 227.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 268.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 567.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 422.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 57.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 517.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 418.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 586.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 595.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 346.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 322.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.10e3T + 9.12e5T^{2} \) |
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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
−17.68010385117086831770481601804, −16.68297179776369890531339719975, −15.16191950508220660408776912611, −13.75781563552186929751488294519, −11.39938497939304475551410015780, −10.55940034165964641102164597423, −8.855824625149566097471981464697, −8.263389515981525643308750721425, −6.21417340157474231380195744645, −1.85619229170449453679309191065,
1.85619229170449453679309191065, 6.21417340157474231380195744645, 8.263389515981525643308750721425, 8.855824625149566097471981464697, 10.55940034165964641102164597423, 11.39938497939304475551410015780, 13.75781563552186929751488294519, 15.16191950508220660408776912611, 16.68297179776369890531339719975, 17.68010385117086831770481601804
