L(s) = 1 | + (−0.182 − 0.983i)2-s + (−0.509 + 0.860i)3-s + (−0.933 + 0.358i)4-s + (0.321 − 0.947i)5-s + (0.939 + 0.343i)6-s + (0.495 − 0.868i)7-s + (0.522 + 0.852i)8-s + (−0.481 − 0.876i)9-s + (−0.989 − 0.143i)10-s + (−0.880 − 0.474i)11-s + (0.166 − 0.986i)12-s + (0.981 + 0.190i)13-s + (−0.944 − 0.328i)14-s + (0.651 + 0.758i)15-s + (0.742 − 0.669i)16-s + (−0.963 − 0.267i)17-s + ⋯ |
L(s) = 1 | + (−0.182 − 0.983i)2-s + (−0.509 + 0.860i)3-s + (−0.933 + 0.358i)4-s + (0.321 − 0.947i)5-s + (0.939 + 0.343i)6-s + (0.495 − 0.868i)7-s + (0.522 + 0.852i)8-s + (−0.481 − 0.876i)9-s + (−0.989 − 0.143i)10-s + (−0.880 − 0.474i)11-s + (0.166 − 0.986i)12-s + (0.981 + 0.190i)13-s + (−0.944 − 0.328i)14-s + (0.651 + 0.758i)15-s + (0.742 − 0.669i)16-s + (−0.963 − 0.267i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9664857361 - 0.6435887490i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9664857361 - 0.6435887490i\) |
\(L(1)\) |
\(\approx\) |
\(0.7372558550 - 0.3464435494i\) |
\(L(1)\) |
\(\approx\) |
\(0.7372558550 - 0.3464435494i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 4729 | \( 1 \) |
good | 2 | \( 1 + (-0.182 - 0.983i)T \) |
| 3 | \( 1 + (-0.509 + 0.860i)T \) |
| 5 | \( 1 + (0.321 - 0.947i)T \) |
| 7 | \( 1 + (0.495 - 0.868i)T \) |
| 11 | \( 1 + (-0.880 - 0.474i)T \) |
| 13 | \( 1 + (0.981 + 0.190i)T \) |
| 17 | \( 1 + (-0.963 - 0.267i)T \) |
| 19 | \( 1 + (0.967 + 0.252i)T \) |
| 23 | \( 1 + (-0.978 + 0.205i)T \) |
| 29 | \( 1 + (-0.182 + 0.983i)T \) |
| 31 | \( 1 + (-0.536 + 0.843i)T \) |
| 37 | \( 1 + (0.229 - 0.973i)T \) |
| 41 | \( 1 + (-0.213 + 0.976i)T \) |
| 43 | \( 1 + (0.803 - 0.595i)T \) |
| 47 | \( 1 + (0.103 + 0.994i)T \) |
| 53 | \( 1 + (0.839 + 0.543i)T \) |
| 59 | \( 1 + (0.995 - 0.0955i)T \) |
| 61 | \( 1 + (0.351 + 0.936i)T \) |
| 67 | \( 1 + (0.997 + 0.0637i)T \) |
| 71 | \( 1 + (-0.0875 + 0.996i)T \) |
| 73 | \( 1 + (0.949 + 0.313i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.563 + 0.826i)T \) |
| 89 | \( 1 + (-0.395 - 0.918i)T \) |
| 97 | \( 1 + (0.495 - 0.868i)T \) |
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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
−18.15517731520220050744924758253, −17.82989544006051360484424433454, −17.19942589853296394925276249372, −16.18934724998001627010847430610, −15.533260059626140229079402088577, −15.1297849980202316882853064878, −14.26704222819736718508625599558, −13.514667465227884060061743862162, −13.26638319443758960902059655376, −12.30164839703859742118667664694, −11.44334365817645625165220372897, −10.89293417114169077312233110721, −10.09703617625105240162881302510, −9.33310970921706323714623296110, −8.29563987032906199408943383782, −7.982746019260105651650659931257, −7.17861419968200396890220065686, −6.53995981206812166675075302056, −5.82978172119670029771205462738, −5.50922638229738381950069732215, −4.60514595409007738136295967334, −3.52513223949508309844907428342, −2.282845989068099539019300620973, −1.951123260131022184088742211371, −0.59374327031880633699607346166,
0.664033856387287599426962994651, 1.26287853982023834867718885329, 2.275374293682839065399271547088, 3.389614385330202742604178524667, 3.99524062755281974797334199978, 4.58552430600758470304285271091, 5.360777017096399778639645402267, 5.768234980454216789209602300872, 7.09961292545554374222744978244, 8.11522663667949062178479397104, 8.65443642777341127491367793265, 9.31200968220364553829238886863, 9.9902444170308130951634307859, 10.68919645194799903133881517489, 11.1206397824876214001395330633, 11.732490173148176826316975996963, 12.601592836043687547848897294985, 13.22855655441833392551962903613, 13.92719145355713036568641804287, 14.34683473826687408561846462299, 15.73878955794993392225316744318, 16.172333746836062143385460429862, 16.656058939741995092825619380959, 17.56035778907061120604386351844, 17.975185989702241542633376335472