L(s) = 1 | − i·3-s + 3i·7-s − 9-s + 5i·13-s + 5·19-s + 3·21-s + 4i·23-s + i·27-s + 4·29-s − 5·31-s − 10i·37-s + 5·39-s − 10·41-s − i·43-s + 2i·47-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.13i·7-s − 0.333·9-s + 1.38i·13-s + 1.14·19-s + 0.654·21-s + 0.834i·23-s + 0.192i·27-s + 0.742·29-s − 0.898·31-s − 1.64i·37-s + 0.800·39-s − 1.56·41-s − 0.152i·43-s + 0.291i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.160431016\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.160431016\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 3iT - 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 14iT - 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 - 5iT - 97T^{2} \) |
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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
−8.606938057412524202999607600907, −7.67477548776952458957959219757, −7.13286337879199020883355931321, −6.34142808508150924037838829011, −5.63779364263336864356753665397, −5.02972946519885782135700275423, −3.96055052278994043085248526918, −3.02840698547953707872664159336, −2.14295431639444958623724078918, −1.37607850092143067310430026516,
0.32081068454276407918212197672, 1.39241911126522342939522615265, 2.92978096099522322815181471280, 3.39543492773350939285457479261, 4.34739751737343956023405624561, 5.03888969184073267711721241618, 5.71731648523789680883129590211, 6.71455144011766870407364801856, 7.31491457990130331438229770146, 8.147241183398007594079077610277