L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.258 + 0.965i)3-s + (−0.866 − 0.499i)4-s + (0.866 + 0.499i)6-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.20 − 0.158i)11-s + (0.707 − 0.707i)12-s + (0.500 + 0.866i)16-s + (0.965 − 0.258i)17-s + (−0.707 + 0.707i)18-s + (1.22 + 1.22i)19-s + (0.158 − 1.20i)22-s + (−0.500 − 0.866i)24-s + (−0.965 − 0.258i)25-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.258 + 0.965i)3-s + (−0.866 − 0.499i)4-s + (0.866 + 0.499i)6-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.20 − 0.158i)11-s + (0.707 − 0.707i)12-s + (0.500 + 0.866i)16-s + (0.965 − 0.258i)17-s + (−0.707 + 0.707i)18-s + (1.22 + 1.22i)19-s + (0.158 − 1.20i)22-s + (−0.500 − 0.866i)24-s + (−0.965 − 0.258i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.059176767\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.059176767\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 + (0.258 - 0.965i)T \) |
| 17 | \( 1 + (-0.965 + 0.258i)T \) |
good | 5 | \( 1 + (0.965 + 0.258i)T^{2} \) |
| 7 | \( 1 + (0.965 - 0.258i)T^{2} \) |
| 11 | \( 1 + (-1.20 + 0.158i)T + (0.965 - 0.258i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 23 | \( 1 + (0.258 - 0.965i)T^{2} \) |
| 29 | \( 1 + (-0.258 - 0.965i)T^{2} \) |
| 31 | \( 1 + (-0.965 - 0.258i)T^{2} \) |
| 37 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + (-0.965 + 0.741i)T + (0.258 - 0.965i)T^{2} \) |
| 43 | \( 1 + (-1.86 - 0.5i)T + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (0.965 - 0.258i)T^{2} \) |
| 67 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (1.83 - 0.758i)T + (0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.965 + 0.258i)T^{2} \) |
| 83 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + (1.25 + 0.965i)T + (0.258 + 0.965i)T^{2} \) |
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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
−9.817039262225021723122822747040, −9.482813714982416356293994911296, −8.617034092769477412091937083376, −7.56416142453260072905423102797, −5.96778173226452283816290081905, −5.63913442958703769281156030194, −4.40400776278342584999541919851, −3.77646176579393268455697080914, −2.93520718686769804224954247384, −1.29341746350696053237959046508,
1.18231830807133550756794244010, 2.91701035474480357953185250192, 4.04919649090401265982845675704, 5.19233679422473653005162465507, 5.94526216928123463177675212173, 6.67855634371846870465125446332, 7.47025166955420285874432785678, 7.959670605897207792145216842347, 9.120615660421147480936207586499, 9.526293705077069010558410399152