| L(s) = 1 | − 6.95i·5-s + 10.6i·7-s − 3.82i·11-s + 12.1·13-s − 10.7i·17-s + 11.6i·19-s + (20.3 + 10.7i)23-s − 23.3·25-s − 33.3·29-s + 32.5·31-s + 73.9·35-s − 22.3i·37-s + 73.9·41-s − 11.6i·43-s + 40.6·47-s + ⋯ |
| L(s) = 1 | − 1.39i·5-s + 1.51i·7-s − 0.347i·11-s + 0.938·13-s − 0.634i·17-s + 0.614i·19-s + (0.883 + 0.468i)23-s − 0.935·25-s − 1.14·29-s + 1.05·31-s + 2.11·35-s − 0.603i·37-s + 1.80·41-s − 0.271i·43-s + 0.864·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.942304393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.942304393\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (-20.3 - 10.7i)T \) |
| good | 5 | \( 1 + 6.95iT - 25T^{2} \) |
| 7 | \( 1 - 10.6iT - 49T^{2} \) |
| 11 | \( 1 + 3.82iT - 121T^{2} \) |
| 13 | \( 1 - 12.1T + 169T^{2} \) |
| 17 | \( 1 + 10.7iT - 289T^{2} \) |
| 19 | \( 1 - 11.6iT - 361T^{2} \) |
| 29 | \( 1 + 33.3T + 841T^{2} \) |
| 31 | \( 1 - 32.5T + 961T^{2} \) |
| 37 | \( 1 + 22.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 73.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 11.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 40.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 15.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 7.32T + 3.48e3T^{2} \) |
| 61 | \( 1 + 88.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 28.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 7.32T + 5.04e3T^{2} \) |
| 73 | \( 1 - 99.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 29.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 147. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 85.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 59.5iT - 9.40e3T^{2} \) |
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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
−9.574077856698765636573370322544, −9.027011061524512659000930134824, −8.525408718690802471323679940114, −7.62160110225507036638196153013, −6.12784341000808780418945226523, −5.55442739033003502806084301815, −4.75395323898343855707860841814, −3.49525924451069851185195799838, −2.15656637948792966637639577002, −0.869985833541960584414805224954,
1.01680713227161311445235449849, 2.62820387350688166207854168233, 3.67330599234656155884752938139, 4.42509840516951675319772588780, 5.94869574706517477788032790007, 6.82622433034959032188497215861, 7.28033944194119699674696127839, 8.223148884857999997107231849982, 9.426165598311864920703862573761, 10.33922712623086743718428430613
