L(s) = 1 | + (0.826 + 0.563i)5-s + (−0.988 + 0.149i)11-s + (−0.365 + 0.930i)13-s + (−0.222 − 0.974i)17-s + 19-s + (0.955 − 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.955 − 0.294i)29-s + (−0.5 − 0.866i)31-s + (−0.222 − 0.974i)37-s + (0.826 + 0.563i)41-s + (−0.826 + 0.563i)43-s + (0.988 − 0.149i)47-s + (0.222 − 0.974i)53-s + (−0.900 − 0.433i)55-s + ⋯ |
L(s) = 1 | + (0.826 + 0.563i)5-s + (−0.988 + 0.149i)11-s + (−0.365 + 0.930i)13-s + (−0.222 − 0.974i)17-s + 19-s + (0.955 − 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.955 − 0.294i)29-s + (−0.5 − 0.866i)31-s + (−0.222 − 0.974i)37-s + (0.826 + 0.563i)41-s + (−0.826 + 0.563i)43-s + (0.988 − 0.149i)47-s + (0.222 − 0.974i)53-s + (−0.900 − 0.433i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.183947701 - 0.3132825078i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.183947701 - 0.3132825078i\) |
\(L(1)\) |
\(\approx\) |
\(1.167939573 + 0.06799350175i\) |
\(L(1)\) |
\(\approx\) |
\(1.167939573 + 0.06799350175i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.826 + 0.563i)T \) |
| 11 | \( 1 + (-0.988 + 0.149i)T \) |
| 13 | \( 1 + (-0.365 + 0.930i)T \) |
| 17 | \( 1 + (-0.222 - 0.974i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.955 - 0.294i)T \) |
| 29 | \( 1 + (-0.955 - 0.294i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.222 - 0.974i)T \) |
| 41 | \( 1 + (0.826 + 0.563i)T \) |
| 43 | \( 1 + (-0.826 + 0.563i)T \) |
| 47 | \( 1 + (0.988 - 0.149i)T \) |
| 53 | \( 1 + (0.222 - 0.974i)T \) |
| 59 | \( 1 + (-0.0747 - 0.997i)T \) |
| 61 | \( 1 + (-0.955 - 0.294i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.222 + 0.974i)T \) |
| 73 | \( 1 + (-0.623 + 0.781i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.365 - 0.930i)T \) |
| 89 | \( 1 + (0.623 - 0.781i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.24774862286814668341291678054, −19.44137179730595310687868940499, −18.40856375881080151338664099868, −17.93274421379365588720439343188, −17.1032423661118818434055862751, −16.581783180755679837430907902517, −15.54715664341550045515818329931, −15.06712154240143165572031253799, −13.96785618259908208740761602163, −13.332982009690591526154061795778, −12.739797107830529511618265822590, −12.06744792445244094756564748658, −10.73130527576551550065205802585, −10.45974506661282554967198739322, −9.43378320255434811995056614342, −8.79655385066352358842096931810, −7.88705397292124224972190420776, −7.16847931835308562818897863744, −5.97374335073939242241347553759, −5.389972100868626887426042684, −4.80152059407953380620578792569, −3.48165099062316057674447782158, −2.681650847864619739101842545039, −1.6716041235520171366296266079, −0.7505650079597498768970507991,
0.49867820279867258034115441737, 1.85129991844534706078952276558, 2.524599986741204073428268137318, 3.34590381948590878070532940596, 4.60633973891529645465590482493, 5.333424746734582374187160480088, 6.09467357431069712637815409680, 7.22173742963452491022566916803, 7.425825886466515550485279316345, 8.83771572714340381941941771984, 9.535903095660855105364687544986, 10.06582600170958527093041708792, 11.13647446926574234989552517266, 11.51872422529408033674512337100, 12.79474403926493890352430137698, 13.30505224087686219991392860409, 14.1380766461338125928786010923, 14.680657962203286398203232892420, 15.60600183862200655774974039586, 16.37091930326336885022660813534, 17.11214348088241856000854130584, 17.95416101012624306443564640624, 18.5240382569544165157077870061, 19.02899520223560143387511950828, 20.204852460089256166044703496872