L(s) = 1 | + (−0.409 − 0.912i)2-s + (−0.999 − 0.0337i)3-s + (−0.664 + 0.747i)4-s + (−0.780 + 0.625i)5-s + (0.378 + 0.925i)6-s + (0.347 + 0.937i)7-s + (0.954 + 0.299i)8-s + (0.997 + 0.0675i)9-s + (0.890 + 0.455i)10-s + (−0.217 + 0.975i)11-s + (0.688 − 0.724i)12-s + (−0.954 + 0.299i)13-s + (0.713 − 0.701i)14-s + (0.801 − 0.598i)15-s + (−0.117 − 0.993i)16-s + (0.979 + 0.201i)17-s + ⋯ |
L(s) = 1 | + (−0.409 − 0.912i)2-s + (−0.999 − 0.0337i)3-s + (−0.664 + 0.747i)4-s + (−0.780 + 0.625i)5-s + (0.378 + 0.925i)6-s + (0.347 + 0.937i)7-s + (0.954 + 0.299i)8-s + (0.997 + 0.0675i)9-s + (0.890 + 0.455i)10-s + (−0.217 + 0.975i)11-s + (0.688 − 0.724i)12-s + (−0.954 + 0.299i)13-s + (0.713 − 0.701i)14-s + (0.801 − 0.598i)15-s + (−0.117 − 0.993i)16-s + (0.979 + 0.201i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.06592582257 + 0.2003843538i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06592582257 + 0.2003843538i\) |
\(L(1)\) |
\(\approx\) |
\(0.4478642162 + 0.009367228089i\) |
\(L(1)\) |
\(\approx\) |
\(0.4478642162 + 0.009367228089i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 373 | \( 1 \) |
good | 2 | \( 1 + (-0.409 - 0.912i)T \) |
| 3 | \( 1 + (-0.999 - 0.0337i)T \) |
| 5 | \( 1 + (-0.780 + 0.625i)T \) |
| 7 | \( 1 + (0.347 + 0.937i)T \) |
| 11 | \( 1 + (-0.217 + 0.975i)T \) |
| 13 | \( 1 + (-0.954 + 0.299i)T \) |
| 17 | \( 1 + (0.979 + 0.201i)T \) |
| 19 | \( 1 + (-0.918 + 0.394i)T \) |
| 23 | \( 1 + (-0.528 - 0.848i)T \) |
| 29 | \( 1 + (0.857 - 0.514i)T \) |
| 31 | \( 1 + (0.688 + 0.724i)T \) |
| 37 | \( 1 + (-0.905 + 0.425i)T \) |
| 41 | \( 1 + (-0.612 + 0.790i)T \) |
| 43 | \( 1 + (0.664 - 0.747i)T \) |
| 47 | \( 1 + (-0.963 + 0.266i)T \) |
| 53 | \( 1 + (-0.997 + 0.0675i)T \) |
| 59 | \( 1 + (0.0168 + 0.999i)T \) |
| 61 | \( 1 + (0.999 + 0.0337i)T \) |
| 67 | \( 1 + (-0.151 - 0.988i)T \) |
| 71 | \( 1 + (-0.315 + 0.948i)T \) |
| 73 | \( 1 + (-0.557 - 0.830i)T \) |
| 79 | \( 1 + (-0.890 - 0.455i)T \) |
| 83 | \( 1 + (0.997 - 0.0675i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.979 - 0.201i)T \) |
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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
−24.04976572119868932967622311534, −23.62491146153198177676005148623, −22.95089163436458957346917333022, −21.87861604207221885148143617078, −20.79027870910971827892729546072, −19.48578890436340406567561167988, −19.00078310881742031242167169742, −17.6166681754755885129749184095, −17.16690137729395385694902525035, −16.331704198890571309384053616242, −15.77703097209558341593763181139, −14.650035419006073424853245634, −13.56495681296364465134522505863, −12.5702005542062540393328557782, −11.465502381823102393659667474521, −10.561644339940934190392091588447, −9.699851895345876512933487872244, −8.299011504624634604123394095474, −7.622298852759608559350953179144, −6.7188665212711825194305664476, −5.47449259598083025690686477517, −4.79548505758146020410823525364, −3.802596770905038307332679931548, −1.20961279250704546621357297660, −0.19302347561520966497583175204,
1.72563858249513088633603639957, 2.79608049919236853894461389392, 4.27790427324980988958652672862, 4.967722121029785654701819377655, 6.4765224495393796828463315071, 7.56744217212634085076274770289, 8.43664790226711920254783042778, 9.96409092829987095625224602696, 10.392307350822435549386377883219, 11.580241421070253044324529510230, 12.20433417538449499550493194357, 12.55124281611801586470302185853, 14.313275034167453829165586969331, 15.223553057363953362915845029876, 16.28492992483383825226556168601, 17.33290159988846577400619480429, 18.01181845782523260029810602544, 18.8955470305083743336436242233, 19.33930478524164580161009328925, 20.72180738597036700219440138716, 21.52719658288948440900915712809, 22.30350911453130503462833524218, 22.97608968167710133177823959588, 23.75436134399307295588714365850, 25.0147684895563994608547607519