| L(s) = 1 | + i·3-s + (2 + i)5-s + i·7-s − 9-s − 4·11-s + 2i·13-s + (−1 + 2i)15-s − 2i·17-s − 2·19-s − 21-s + 6i·23-s + (3 + 4i)25-s − i·27-s − 6·29-s − 6·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.894 + 0.447i)5-s + 0.377i·7-s − 0.333·9-s − 1.20·11-s + 0.554i·13-s + (−0.258 + 0.516i)15-s − 0.485i·17-s − 0.458·19-s − 0.218·21-s + 1.25i·23-s + (0.600 + 0.800i)25-s − 0.192i·27-s − 1.11·29-s − 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.184505347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.184505347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2 - i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.473085835099832135961826760589, −9.309383403710638666001786121605, −8.129618539359861725128170546132, −7.31389556620405462878996554263, −6.35471301963850703238367892455, −5.48754715163448808142478910701, −5.00808821292139965791964853524, −3.70456521129806073514636610767, −2.74431134969039036229484213345, −1.84652139555756762753248856884,
0.41002701862452815638068003388, 1.84559586549780823792609125113, 2.64614280749033305500312946816, 3.96093171659562900994942581441, 5.16279447382091566237105160711, 5.68340562035333934168063884294, 6.59041345150838864876647244373, 7.46076405834372518269636748419, 8.238221615209891668067861805353, 8.892317637351689119864639239854
