| L(s) = 1 | + 1.73i·3-s + 4.28i·7-s − 2.99·9-s − 11.2i·11-s − 5.41·13-s + 1.41·17-s − 8.56i·19-s − 7.41·21-s + 24.0i·23-s − 5.19i·27-s − 25.4·29-s − 50.1i·31-s + 19.4·33-s − 60.2·37-s − 9.38i·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.611i·7-s − 0.333·9-s − 1.01i·11-s − 0.416·13-s + 0.0833·17-s − 0.450i·19-s − 0.353·21-s + 1.04i·23-s − 0.192i·27-s − 0.876·29-s − 1.61i·31-s + 0.588·33-s − 1.62·37-s − 0.240i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8445051906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8445051906\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.28iT - 49T^{2} \) |
| 11 | \( 1 + 11.2iT - 121T^{2} \) |
| 13 | \( 1 + 5.41T + 169T^{2} \) |
| 17 | \( 1 - 1.41T + 289T^{2} \) |
| 19 | \( 1 + 8.56iT - 361T^{2} \) |
| 23 | \( 1 - 24.0iT - 529T^{2} \) |
| 29 | \( 1 + 25.4T + 841T^{2} \) |
| 31 | \( 1 + 50.1iT - 961T^{2} \) |
| 37 | \( 1 + 60.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 30T + 1.68e3T^{2} \) |
| 43 | \( 1 + 37.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 39.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 16.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 81.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 94.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 129. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 24.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 91.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 88.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 20.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 14T + 9.40e3T^{2} \) |
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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
−9.269721944020449166060534615509, −8.730646149533927313493730461190, −7.82213139933745807707595943561, −6.84968775389395987946666560991, −5.67513229124836499782507366236, −5.31599861232935864517128127414, −4.01008728866875793926958782787, −3.18803154321025364964927329695, −2.03241615847625030520390729933, −0.25153955402928019706668075624,
1.28323575969235786946082450034, 2.35342785543453239058044400281, 3.60027561389014095386386829993, 4.64608477431252611883254491615, 5.51486749545344234279132264358, 6.80214274287827010795820581186, 7.08450167839768146177546318104, 8.062985592247398087833979851419, 8.837566908402630152358652885219, 9.914444770910279530153739276463
