L(s) = 1 | + (−0.524 − 0.851i)2-s + (−0.778 − 0.628i)3-s + (−0.450 + 0.892i)4-s + (0.660 − 0.750i)5-s + (−0.127 + 0.991i)6-s + (0.985 + 0.169i)7-s + (0.996 − 0.0848i)8-s + (0.210 + 0.977i)9-s + (−0.985 − 0.169i)10-s + (0.450 + 0.892i)11-s + (0.911 − 0.411i)12-s + (0.828 + 0.559i)13-s + (−0.372 − 0.927i)14-s + (−0.985 + 0.169i)15-s + (−0.594 − 0.803i)16-s + (0.778 + 0.628i)17-s + ⋯ |
L(s) = 1 | + (−0.524 − 0.851i)2-s + (−0.778 − 0.628i)3-s + (−0.450 + 0.892i)4-s + (0.660 − 0.750i)5-s + (−0.127 + 0.991i)6-s + (0.985 + 0.169i)7-s + (0.996 − 0.0848i)8-s + (0.210 + 0.977i)9-s + (−0.985 − 0.169i)10-s + (0.450 + 0.892i)11-s + (0.911 − 0.411i)12-s + (0.828 + 0.559i)13-s + (−0.372 − 0.927i)14-s + (−0.985 + 0.169i)15-s + (−0.594 − 0.803i)16-s + (0.778 + 0.628i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6361606002 - 0.5322992051i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6361606002 - 0.5322992051i\) |
\(L(1)\) |
\(\approx\) |
\(0.7019891431 - 0.4101529708i\) |
\(L(1)\) |
\(\approx\) |
\(0.7019891431 - 0.4101529708i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 149 | \( 1 \) |
good | 2 | \( 1 + (-0.524 - 0.851i)T \) |
| 3 | \( 1 + (-0.778 - 0.628i)T \) |
| 5 | \( 1 + (0.660 - 0.750i)T \) |
| 7 | \( 1 + (0.985 + 0.169i)T \) |
| 11 | \( 1 + (0.450 + 0.892i)T \) |
| 13 | \( 1 + (0.828 + 0.559i)T \) |
| 17 | \( 1 + (0.778 + 0.628i)T \) |
| 19 | \( 1 + (-0.721 - 0.691i)T \) |
| 23 | \( 1 + (0.828 - 0.559i)T \) |
| 29 | \( 1 + (-0.967 + 0.251i)T \) |
| 31 | \( 1 + (-0.828 + 0.559i)T \) |
| 37 | \( 1 + (-0.450 - 0.892i)T \) |
| 41 | \( 1 + (0.594 + 0.803i)T \) |
| 43 | \( 1 + (-0.873 - 0.487i)T \) |
| 47 | \( 1 + (-0.721 + 0.691i)T \) |
| 53 | \( 1 + (0.873 - 0.487i)T \) |
| 59 | \( 1 + (-0.0424 - 0.999i)T \) |
| 61 | \( 1 + (0.524 + 0.851i)T \) |
| 67 | \( 1 + (0.372 - 0.927i)T \) |
| 71 | \( 1 + (-0.660 + 0.750i)T \) |
| 73 | \( 1 + (0.942 - 0.333i)T \) |
| 79 | \( 1 + (-0.942 - 0.333i)T \) |
| 83 | \( 1 + (-0.942 + 0.333i)T \) |
| 89 | \( 1 + (0.292 - 0.956i)T \) |
| 97 | \( 1 + (-0.873 + 0.487i)T \) |
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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
−27.77389625289796144057986613573, −27.46596035377685179195012784380, −26.49298487299478088885057863736, −25.50572592584154833583066376507, −24.501099102531285263218971437357, −23.36928200461052289646194129195, −22.67866260344927110198794695668, −21.540056600224047992052925517589, −20.64668553122737327097742940413, −18.81043635628944951243400703081, −18.17818602009325556432682593856, −17.17419091392349304409405643077, −16.56657830193491672108635620608, −15.21229467688891910381425576919, −14.52882038029331498989213537640, −13.44590568778272631589934466275, −11.376998795180431437734869582712, −10.74219949108667161645733443466, −9.73087364072330598642687891791, −8.537953958322415367284614422698, −7.16628880505679663847776710909, −5.93874012175550956723507930781, −5.3451801688726313623587192026, −3.721324296868165316816718384668, −1.31176685782911552483512238153,
1.33467405867698974522236182624, 1.991684419733439723768667155855, 4.30096265923322393113081565966, 5.330876555515790627204818570704, 6.89666730216160336407472469800, 8.23936889980860148742256187923, 9.18293399749586440500745830346, 10.56836630541283177420416183530, 11.438316779239329161277829932114, 12.489937682770936159550706923898, 13.13151664438662690911108924081, 14.453676679679862530438383855225, 16.44524011639542785248212684864, 17.20988906944645556137645581644, 17.878694692132338581875612155699, 18.7620474006621782271867582081, 19.965511573840063763740517943211, 21.07062617464574294272314376615, 21.641651769475016313248958121412, 22.951513316683705291137401978274, 23.94320625580327649178942050759, 25.049091690804792337880962221610, 25.8710064958330595536172207024, 27.507958288813026549593053264085, 28.15094228332871383913892523324