L(s) = 1 | − 0.550i·11-s + 7.89·17-s − 6.34i·19-s + 5·25-s − 6.79·41-s + 12.3i·43-s − 7·49-s − 15.2i·59-s + 0.348i·67-s − 15.6·73-s − 18i·83-s + 18·89-s + 9.69·97-s + 14.1i·107-s + 18·113-s + ⋯ |
L(s) = 1 | − 0.165i·11-s + 1.91·17-s − 1.45i·19-s + 25-s − 1.06·41-s + 1.88i·43-s − 49-s − 1.98i·59-s + 0.0425i·67-s − 1.83·73-s − 1.97i·83-s + 1.90·89-s + 0.984·97-s + 1.36i·107-s + 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.976593921\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.976593921\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 0.550iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 7.89T + 17T^{2} \) |
| 19 | \( 1 + 6.34iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 6.79T + 41T^{2} \) |
| 43 | \( 1 - 12.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 15.2iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 0.348iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 18iT - 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 - 9.69T + 97T^{2} \) |
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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
−8.063798025436533954337913475067, −7.46776100489292448906640457552, −6.67754799227442316497968354130, −6.00796228886776588811722399812, −5.07702938313688218143624007847, −4.63820419933221841008478642101, −3.34337803067026518924717734309, −2.98172640407807164500949864115, −1.67343209789532726374846175922, −0.63837602547448816193466052473,
0.997806460989646868888582719766, 1.91629668148390267814323180266, 3.13965058866266768045650602992, 3.66085489897178372210763034627, 4.67435000513673028117727693744, 5.51212888755915875708811804978, 5.98068714180352262520574206623, 7.00716824502958814139309809091, 7.55807688866920765192140426851, 8.298980743647979808943865653865