L(s) = 1 | + (0.958 − 0.285i)2-s + (−0.443 − 0.896i)3-s + (0.836 − 0.548i)4-s + (0.715 − 0.698i)5-s + (−0.681 − 0.732i)6-s + (−0.906 − 0.421i)7-s + (0.644 − 0.764i)8-s + (−0.607 + 0.794i)9-s + (0.485 − 0.873i)10-s + (−0.861 + 0.506i)11-s + (−0.861 − 0.506i)12-s + (0.485 + 0.873i)13-s + (−0.989 − 0.144i)14-s + (−0.943 − 0.331i)15-s + (0.399 − 0.916i)16-s + (0.995 + 0.0965i)17-s + ⋯ |
L(s) = 1 | + (0.958 − 0.285i)2-s + (−0.443 − 0.896i)3-s + (0.836 − 0.548i)4-s + (0.715 − 0.698i)5-s + (−0.681 − 0.732i)6-s + (−0.906 − 0.421i)7-s + (0.644 − 0.764i)8-s + (−0.607 + 0.794i)9-s + (0.485 − 0.873i)10-s + (−0.861 + 0.506i)11-s + (−0.861 − 0.506i)12-s + (0.485 + 0.873i)13-s + (−0.989 − 0.144i)14-s + (−0.943 − 0.331i)15-s + (0.399 − 0.916i)16-s + (0.995 + 0.0965i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.046030830 - 1.255951932i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.046030830 - 1.255951932i\) |
\(L(1)\) |
\(\approx\) |
\(1.296203424 - 0.8502102962i\) |
\(L(1)\) |
\(\approx\) |
\(1.296203424 - 0.8502102962i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 131 | \( 1 \) |
good | 2 | \( 1 + (0.958 - 0.285i)T \) |
| 3 | \( 1 + (-0.443 - 0.896i)T \) |
| 5 | \( 1 + (0.715 - 0.698i)T \) |
| 7 | \( 1 + (-0.906 - 0.421i)T \) |
| 11 | \( 1 + (-0.861 + 0.506i)T \) |
| 13 | \( 1 + (0.485 + 0.873i)T \) |
| 17 | \( 1 + (0.995 + 0.0965i)T \) |
| 19 | \( 1 + (-0.748 + 0.663i)T \) |
| 23 | \( 1 + (0.926 - 0.377i)T \) |
| 29 | \( 1 + (-0.607 - 0.794i)T \) |
| 31 | \( 1 + (-0.527 - 0.849i)T \) |
| 37 | \( 1 + (0.779 + 0.626i)T \) |
| 41 | \( 1 + (0.399 + 0.916i)T \) |
| 43 | \( 1 + (-0.168 + 0.985i)T \) |
| 47 | \( 1 + (0.568 + 0.822i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.215 + 0.976i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 + (-0.168 - 0.985i)T \) |
| 71 | \( 1 + (-0.970 + 0.239i)T \) |
| 73 | \( 1 + (-0.809 - 0.587i)T \) |
| 79 | \( 1 + (0.885 + 0.464i)T \) |
| 83 | \( 1 + (-0.998 + 0.0483i)T \) |
| 89 | \( 1 + (0.309 + 0.951i)T \) |
| 97 | \( 1 + (0.981 - 0.192i)T \) |
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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
−29.21993872047512368695752525902, −28.16339186057106185222767387460, −26.707854557616097920598245950829, −25.7325991766003405488080771631, −25.28841428635499236306718870080, −23.495833152403400878309436016288, −22.91356328213393209353782898114, −21.85301919222097053274915033432, −21.43738517180715832664394039009, −20.33422688936059002968675378440, −18.76052367735828523681639349904, −17.458361565005652377213907659352, −16.38040103109688818172447987506, −15.51919877097515742239913106063, −14.7210485965859328519953037572, −13.432914771218501806749415463, −12.51867283372132473821632137423, −10.98182144902169322588632309056, −10.388259792191617289526496085379, −8.92049356262171042474436899472, −7.0930456019951631031879803089, −5.81266795943448966064358153898, −5.38671402864911866162179285495, −3.502959589797792426645804751, −2.793928062074951632150877194426,
1.33460853599944136407992667848, 2.61408396982158944363446314354, 4.39104061568446600953396732161, 5.71535292249164807249273448974, 6.44559579292465289117468625166, 7.739880802783920419584343332866, 9.62450169777391027709943043513, 10.73439504095040279492107559400, 12.07746615060332419776293396338, 13.04817817851802640568100109750, 13.34260748788109153015635688747, 14.66818958095825221260384434012, 16.377994077096175736259721321474, 16.82694402432905601226400841835, 18.47994491113750296726024143475, 19.3232504517991569769920331481, 20.56452206404945705054941216201, 21.315168170344740359628906313160, 22.68203441578427376142794625479, 23.35324054336706038104093967831, 24.135533291646665720532222098753, 25.30019419191056001055925844623, 25.840424799272518406151535676039, 28.07164727382379647034380486519, 28.797143893541040314793621471420