L(s) = 1 | + (0.526 − 0.850i)2-s + (0.403 + 0.914i)3-s + (−0.445 − 0.895i)4-s + (0.228 − 0.973i)5-s + (0.990 + 0.138i)6-s + (−0.183 − 0.982i)7-s + (−0.995 − 0.0922i)8-s + (−0.673 + 0.739i)9-s + (−0.707 − 0.707i)10-s + (−0.895 + 0.445i)11-s + (0.638 − 0.769i)12-s + (−0.824 − 0.565i)13-s + (−0.932 − 0.361i)14-s + (0.982 − 0.183i)15-s + (−0.602 + 0.798i)16-s + (0.995 − 0.0922i)17-s + ⋯ |
L(s) = 1 | + (0.526 − 0.850i)2-s + (0.403 + 0.914i)3-s + (−0.445 − 0.895i)4-s + (0.228 − 0.973i)5-s + (0.990 + 0.138i)6-s + (−0.183 − 0.982i)7-s + (−0.995 − 0.0922i)8-s + (−0.673 + 0.739i)9-s + (−0.707 − 0.707i)10-s + (−0.895 + 0.445i)11-s + (0.638 − 0.769i)12-s + (−0.824 − 0.565i)13-s + (−0.932 − 0.361i)14-s + (0.982 − 0.183i)15-s + (−0.602 + 0.798i)16-s + (0.995 − 0.0922i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08736914270 - 1.418651269i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08736914270 - 1.418651269i\) |
\(L(1)\) |
\(\approx\) |
\(0.9692357912 - 0.6984880333i\) |
\(L(1)\) |
\(\approx\) |
\(0.9692357912 - 0.6984880333i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 137 | \( 1 \) |
good | 2 | \( 1 + (0.526 - 0.850i)T \) |
| 3 | \( 1 + (0.403 + 0.914i)T \) |
| 5 | \( 1 + (0.228 - 0.973i)T \) |
| 7 | \( 1 + (-0.183 - 0.982i)T \) |
| 11 | \( 1 + (-0.895 + 0.445i)T \) |
| 13 | \( 1 + (-0.824 - 0.565i)T \) |
| 17 | \( 1 + (0.995 - 0.0922i)T \) |
| 19 | \( 1 + (-0.961 + 0.273i)T \) |
| 23 | \( 1 + (0.990 - 0.138i)T \) |
| 29 | \( 1 + (-0.138 - 0.990i)T \) |
| 31 | \( 1 + (-0.873 + 0.486i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.707 - 0.707i)T \) |
| 43 | \( 1 + (0.486 - 0.873i)T \) |
| 47 | \( 1 + (-0.998 + 0.0461i)T \) |
| 53 | \( 1 + (0.873 + 0.486i)T \) |
| 59 | \( 1 + (0.739 + 0.673i)T \) |
| 61 | \( 1 + (-0.673 - 0.739i)T \) |
| 67 | \( 1 + (0.565 - 0.824i)T \) |
| 71 | \( 1 + (0.948 - 0.317i)T \) |
| 73 | \( 1 + (-0.982 - 0.183i)T \) |
| 79 | \( 1 + (0.914 + 0.403i)T \) |
| 83 | \( 1 + (0.769 - 0.638i)T \) |
| 89 | \( 1 + (0.973 + 0.228i)T \) |
| 97 | \( 1 + (0.317 - 0.948i)T \) |
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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
−29.17251411262847640786598174058, −27.44562671495873873263365848934, −26.12702562697226387012370507426, −25.74888602532052715917885428350, −24.77876049614216864522090798484, −23.823883171658978053527069400352, −23.00072485800075661989470874681, −21.83260536028855419100794158532, −21.12499589333011549952171256556, −19.19505812422156871512458787169, −18.59696449925716366343752992577, −17.70416904095447210487624032613, −16.44539681706944609944184539366, −14.90357211482693160671772192197, −14.66477785009891207653973215416, −13.3306868197994596633592374470, −12.54973820473562165423925848424, −11.36112284548726609149193829145, −9.472584818697743441568586506134, −8.28671917157970244342865347843, −7.25767050946062155903836736665, −6.329844422468070275816392876547, −5.315825266568793740447058577024, −3.23111168118396461603912824119, −2.41261463893637737258098290093,
0.414491061792006381042500875492, 2.26560879794932028201577873404, 3.67557796547231394388454024762, 4.7173275270398080191212039541, 5.52348799932273566451852901469, 7.75599761028762290494807115710, 9.16242632165054690247616979317, 10.09584133739540751597746873852, 10.7492610819190163797218685590, 12.41242014770011684676815945536, 13.189679182879150680365037626259, 14.278868187870561206493625587926, 15.275761713946710097575367916686, 16.51826350575890252738657371604, 17.48073530577482687914233553533, 19.226595984602729063921537682823, 20.05104233163984721092020123372, 20.875498993343863251626403629967, 21.32005245397974884515919295836, 22.76051107179576768236560132024, 23.42503339734611405592633214915, 24.72932795536113592644331490729, 25.89589963102189737450659273824, 27.142887637867535083034581353232, 27.76476815194577434139937914178