| L(s) = 1 | − i·5-s − 2i·11-s − 6·13-s + (4 + i)17-s − 4·19-s + 4i·23-s − 25-s + 6i·29-s + 4i·31-s + 2i·37-s + 2i·41-s − 6·43-s + 8·47-s + 7·49-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s − 0.603i·11-s − 1.66·13-s + (0.970 + 0.242i)17-s − 0.917·19-s + 0.834i·23-s − 0.200·25-s + 1.11i·29-s + 0.718i·31-s + 0.328i·37-s + 0.312i·41-s − 0.914·43-s + 1.16·47-s + 49-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.091950163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.091950163\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 17 | \( 1 + (-4 - i)T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 4iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 16iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 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
−8.791568689367181094141614660416, −8.195816435542701799757142060180, −7.36005053705289716326465203256, −6.73689018992750745914477850111, −5.58450885442087520386347686294, −5.18632672804139460382341327621, −4.20111526836897403258992371067, −3.27552179122973168829686010993, −2.29521834589179403903933240294, −1.09749779004981625284925068080,
0.36675508853442505583309345433, 2.11453457722354219458064830463, 2.65205828795735911124663029808, 3.89045262350437305922915372058, 4.64018981373690646763696482465, 5.47472734872206944595525085934, 6.34821923040242409241198327166, 7.19276937923901728618521857399, 7.62848002425321955041862281429, 8.498640853374975959314260565003
