| L(s) = 1 | − 4.18·5-s + 2.58i·7-s − 6.38i·11-s − 11.5·13-s + 7.58·17-s − 0.0311i·19-s + 7.52i·23-s − 7.52·25-s + 13.5·29-s + 29.7i·31-s − 10.8i·35-s − 57.4·37-s + 27.1·41-s − 77.8i·43-s − 43.8i·47-s + ⋯ |
| L(s) = 1 | − 0.836·5-s + 0.369i·7-s − 0.580i·11-s − 0.891·13-s + 0.446·17-s − 0.00163i·19-s + 0.327i·23-s − 0.300·25-s + 0.467·29-s + 0.960i·31-s − 0.308i·35-s − 1.55·37-s + 0.662·41-s − 1.81i·43-s − 0.932i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.308625494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.308625494\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4.18T + 25T^{2} \) |
| 7 | \( 1 - 2.58iT - 49T^{2} \) |
| 11 | \( 1 + 6.38iT - 121T^{2} \) |
| 13 | \( 1 + 11.5T + 169T^{2} \) |
| 17 | \( 1 - 7.58T + 289T^{2} \) |
| 19 | \( 1 + 0.0311iT - 361T^{2} \) |
| 23 | \( 1 - 7.52iT - 529T^{2} \) |
| 29 | \( 1 - 13.5T + 841T^{2} \) |
| 31 | \( 1 - 29.7iT - 961T^{2} \) |
| 37 | \( 1 + 57.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 27.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 77.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 43.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 87.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 67.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 31.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 23.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 50.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 70.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 67.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 8.55iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 5.22T + 7.92e3T^{2} \) |
| 97 | \( 1 - 120.T + 9.40e3T^{2} \) |
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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
−8.890230779978914327846052871745, −8.478486380187281267775107482873, −7.45580390032068586205049263615, −6.99171602665362734538928636235, −5.75358285351407479525986988149, −5.12425659513023933561571848613, −4.01480965648033634482974737862, −3.25225893528737642206647069484, −2.13520208271827569384029951092, −0.60773985277271322279339550719,
0.62431410702845634733201125303, 2.09585155379554902345656075088, 3.23097720676597104339425997038, 4.20807348016063581573665505052, 4.83152817080182734561097372619, 5.92583480146581123814095612733, 6.95610293379713660466030074841, 7.58598166180848206342136834205, 8.138747266679605980084539092347, 9.200898188554328933240600283093
