L(s) = 1 | − i·3-s + i·5-s − 9-s − 4i·11-s − 6i·13-s + 15-s − 6·17-s + 4i·19-s − 25-s + i·27-s + 2i·29-s − 8·31-s − 4·33-s − 2i·37-s − 6·39-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.447i·5-s − 0.333·9-s − 1.20i·11-s − 1.66i·13-s + 0.258·15-s − 1.45·17-s + 0.917i·19-s − 0.200·25-s + 0.192i·27-s + 0.371i·29-s − 1.43·31-s − 0.696·33-s − 0.328i·37-s − 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + 14iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−7.912796599625954559781492977497, −7.45780800984041549694161985168, −6.42206554088495430335224426829, −5.93266186724602601673909692579, −5.25105498951903355415298803549, −4.02029321570309562791742492301, −3.17169513287499185040718865944, −2.48764007961521057354294396647, −1.22354189487460569191438815017, 0,
1.79158125388477419855318330873, 2.42346376613088842617251768509, 3.88078615245765628348069932634, 4.41252551121001951351896294295, 4.94089114047611548370176630028, 5.93975246916964310309873161845, 6.97877706390390097886219572548, 7.18617849035328307979332807699, 8.500325317326143307596410904104