L(s) = 1 | + i·7-s − 9-s − 11-s + i·17-s − 19-s + 2i·23-s − i·43-s + i·47-s − 61-s − i·63-s − i·73-s − i·77-s + 81-s + 2i·83-s + 99-s + ⋯ |
L(s) = 1 | + i·7-s − 9-s − 11-s + i·17-s − 19-s + 2i·23-s − i·43-s + i·47-s − 61-s − i·63-s − i·73-s − i·77-s + 81-s + 2i·83-s + 99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6738358016\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6738358016\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + T^{2} \) |
| 7 | \( 1 - iT - T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 - 2iT - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 - iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 2iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.509947449414602419106130512195, −8.809579887284687448615259578896, −8.193609026008999431328878211945, −7.48196833966580025376413301999, −6.22018702024616175722806379509, −5.70871104286145118615489737517, −5.01508359892013033273867357364, −3.73575783513243868403494954662, −2.78733877447800069076163046380, −1.90230738564628567184693902590,
0.45808780948722940640938766709, 2.31783756224388070910610549951, 3.10376437912904939628873921552, 4.32457160612193697710962987326, 4.98472888269810639769922012330, 6.02048931963967397356330763431, 6.79472724693543134209112575982, 7.63875873177797577440868630389, 8.364594750470749039077062346468, 9.013980968579495021598026174935