| L(s) = 1 | − 1.73i·3-s + 3.46i·7-s + 6·9-s + 12.1i·11-s − 16·13-s − 3·17-s − 22.5i·19-s + 5.99·21-s + 45.0i·23-s − 25.9i·27-s − 12·29-s − 31.1i·31-s + 21·33-s − 50·37-s + 27.7i·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.494i·7-s + 0.666·9-s + 1.10i·11-s − 1.23·13-s − 0.176·17-s − 1.18i·19-s + 0.285·21-s + 1.95i·23-s − 0.962i·27-s − 0.413·29-s − 1.00i·31-s + 0.636·33-s − 1.35·37-s + 0.710i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1573694696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1573694696\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.73iT - 9T^{2} \) |
| 7 | \( 1 - 3.46iT - 49T^{2} \) |
| 11 | \( 1 - 12.1iT - 121T^{2} \) |
| 13 | \( 1 + 16T + 169T^{2} \) |
| 17 | \( 1 + 3T + 289T^{2} \) |
| 19 | \( 1 + 22.5iT - 361T^{2} \) |
| 23 | \( 1 - 45.0iT - 529T^{2} \) |
| 29 | \( 1 + 12T + 841T^{2} \) |
| 31 | \( 1 + 31.1iT - 961T^{2} \) |
| 37 | \( 1 + 50T + 1.36e3T^{2} \) |
| 41 | \( 1 + 63T + 1.68e3T^{2} \) |
| 43 | \( 1 + 62.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 13.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 18T + 2.80e3T^{2} \) |
| 59 | \( 1 + 55.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 26T + 3.72e3T^{2} \) |
| 67 | \( 1 + 32.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 27.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 17T + 5.32e3T^{2} \) |
| 79 | \( 1 + 86.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 98.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 99T + 7.92e3T^{2} \) |
| 97 | \( 1 + 134T + 9.40e3T^{2} \) |
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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
−8.948185414455149662049803054903, −7.74136421563971969886142490643, −7.23543663674697051129349853734, −6.69068241294440741920944689281, −5.39374842613718966336911457352, −4.81441132586502868886550586286, −3.68859628094364992245539307756, −2.34588756908464034443033359910, −1.70240331372384513510989926870, −0.04065422243984847380905482747,
1.41262781538728584172551223853, 2.81686273743556262884457729373, 3.79115390364140605183713509408, 4.58737684302068890513168486462, 5.37267451626093133389713609970, 6.48280091495778661299841004614, 7.16964126306109263306982332443, 8.127272651559113617111086629223, 8.823322214875671334831561412707, 9.772709022168780423789410179812
