| L(s) = 1 | − 2.82i·5-s − 3·7-s − 19.7i·11-s − 7·13-s + 14.1i·17-s + 19·19-s + 2.82i·23-s + 17·25-s − 50.9i·29-s − 10·31-s + 8.48i·35-s − 63·37-s − 50.9i·41-s + 50·43-s + 42.4i·47-s + ⋯ |
| L(s) = 1 | − 0.565i·5-s − 0.428·7-s − 1.79i·11-s − 0.538·13-s + 0.831i·17-s + 19-s + 0.122i·23-s + 0.680·25-s − 1.75i·29-s − 0.322·31-s + 0.242i·35-s − 1.70·37-s − 1.24i·41-s + 1.16·43-s + 0.902i·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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6694134157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6694134157\) |
| \(L(2)\) |
|
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 | \( 1 + 2.82iT - 25T^{2} \) |
| 7 | \( 1 + 3T + 49T^{2} \) |
| 11 | \( 1 + 19.7iT - 121T^{2} \) |
| 13 | \( 1 + 7T + 169T^{2} \) |
| 17 | \( 1 - 14.1iT - 289T^{2} \) |
| 19 | \( 1 - 19T + 361T^{2} \) |
| 23 | \( 1 - 2.82iT - 529T^{2} \) |
| 29 | \( 1 + 50.9iT - 841T^{2} \) |
| 31 | \( 1 + 10T + 961T^{2} \) |
| 37 | \( 1 + 63T + 1.36e3T^{2} \) |
| 41 | \( 1 + 50.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 50T + 1.84e3T^{2} \) |
| 47 | \( 1 - 42.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 73.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 98.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 79T + 3.72e3T^{2} \) |
| 67 | \( 1 + 77T + 4.48e3T^{2} \) |
| 71 | \( 1 + 79.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 17T + 5.32e3T^{2} \) |
| 79 | \( 1 + 11T + 6.24e3T^{2} \) |
| 83 | \( 1 + 39.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 42.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 97T + 9.40e3T^{2} \) |
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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
−8.873526510634559285187004139958, −8.004509025735225565649891014709, −7.25918974661838388702913286848, −6.03407169529068706691527801377, −5.71346483663472731883600876694, −4.58220524027678251424301708135, −3.58068222399249477028534359834, −2.76977663610031165753430039870, −1.27699207521254942367360112667, −0.18143057425901219768511046866,
1.56380979525909008909441212196, 2.69079934919382016198610088390, 3.49153400713878681013024899534, 4.78861503085975243198115570516, 5.24101747584989622755903109572, 6.71173765930580505808652560193, 7.02239803414628483129318800603, 7.70212903960657862630179259671, 8.928847668528753553392340133524, 9.633945978649665633925665686847
