| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − i·8-s − 9-s + 4·11-s − i·12-s − 2i·13-s + 16-s + 2i·17-s − i·18-s + 19-s + 4i·22-s − 4i·23-s + 24-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + 1.20·11-s − 0.288i·12-s − 0.554i·13-s + 0.250·16-s + 0.485i·17-s − 0.235i·18-s + 0.229·19-s + 0.852i·22-s − 0.834i·23-s + 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.810139057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.810139057\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.833965476724837117413008451687, −8.267413896694576318783735522270, −7.35752364706309258162195544685, −6.60228428996643886699393067599, −5.84528558062778653975451560080, −5.17264045444295940907439437345, −4.10370906250141825011030795705, −3.70210682105875246603949897418, −2.35682492344967270602662253752, −0.804953668745430946328674224290,
0.924079573645545502987278360893, 1.80072908839523862312797285160, 2.83315462286494875820042924471, 3.79421947765328079982345185482, 4.54228949037610847675954638480, 5.61592551294927645941695704551, 6.36182086690218974639245422670, 7.21758120764841977917678759360, 7.86037906367919305850001206604, 8.937411227781701973129571562601
