| L(s) = 1 | − 2.37·2-s + 3·3-s − 2.37·4-s + 19.4·5-s − 7.11·6-s + 6.74·7-s + 24.6·8-s + 9·9-s − 46.2·10-s + 11·11-s − 7.11·12-s − 60.9·13-s − 16·14-s + 58.4·15-s − 39.3·16-s − 99.1·17-s − 21.3·18-s + 24.7·19-s − 46.2·20-s + 20.2·21-s − 26.0·22-s + 112·23-s + 73.8·24-s + 254.·25-s + 144.·26-s + 27·27-s − 16·28-s + ⋯ |
| L(s) = 1 | − 0.838·2-s + 0.577·3-s − 0.296·4-s + 1.74·5-s − 0.484·6-s + 0.364·7-s + 1.08·8-s + 0.333·9-s − 1.46·10-s + 0.301·11-s − 0.171·12-s − 1.30·13-s − 0.305·14-s + 1.00·15-s − 0.615·16-s − 1.41·17-s − 0.279·18-s + 0.298·19-s − 0.516·20-s + 0.210·21-s − 0.252·22-s + 1.01·23-s + 0.627·24-s + 2.03·25-s + 1.09·26-s + 0.192·27-s − 0.107·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.106187051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.106187051\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 2.37T + 8T^{2} \) |
| 5 | \( 1 - 19.4T + 125T^{2} \) |
| 7 | \( 1 - 6.74T + 343T^{2} \) |
| 13 | \( 1 + 60.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 99.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 24.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 112T + 1.21e4T^{2} \) |
| 29 | \( 1 + 21.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 318.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 252.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 214.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 105.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 325.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 196T + 2.05e5T^{2} \) |
| 61 | \( 1 + 402.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 27.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 300.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 427.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 97.5T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 463.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.79e3T + 9.12e5T^{2} \) |
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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
−16.77078261848685652141940426447, −14.79029922420424043623765682971, −13.83824080161561347647576265176, −12.94652457996615243548870646512, −10.67719953626078168455926542217, −9.522932689938979383729287264358, −8.904018914491728706386586193173, −7.10836220703054412540017612900, −5.03136440772887571564301745924, −1.97220520084630365736708225749,
1.97220520084630365736708225749, 5.03136440772887571564301745924, 7.10836220703054412540017612900, 8.904018914491728706386586193173, 9.522932689938979383729287264358, 10.67719953626078168455926542217, 12.94652457996615243548870646512, 13.83824080161561347647576265176, 14.79029922420424043623765682971, 16.77078261848685652141940426447
