L(s) = 1 | + (0.299 + 0.954i)2-s + (−0.556 − 0.831i)3-s + (−0.820 + 0.571i)4-s + (0.503 + 0.863i)5-s + (0.626 − 0.779i)6-s + (−0.860 + 0.508i)7-s + (−0.791 − 0.611i)8-s + (−0.381 + 0.924i)9-s + (−0.673 + 0.739i)10-s + (−0.972 − 0.233i)11-s + (0.931 + 0.363i)12-s + (−0.966 − 0.257i)13-s + (−0.743 − 0.668i)14-s + (0.437 − 0.899i)15-s + (0.346 − 0.938i)16-s + (0.392 − 0.919i)17-s + ⋯ |
L(s) = 1 | + (0.299 + 0.954i)2-s + (−0.556 − 0.831i)3-s + (−0.820 + 0.571i)4-s + (0.503 + 0.863i)5-s + (0.626 − 0.779i)6-s + (−0.860 + 0.508i)7-s + (−0.791 − 0.611i)8-s + (−0.381 + 0.924i)9-s + (−0.673 + 0.739i)10-s + (−0.972 − 0.233i)11-s + (0.931 + 0.363i)12-s + (−0.966 − 0.257i)13-s + (−0.743 − 0.668i)14-s + (0.437 − 0.899i)15-s + (0.346 − 0.938i)16-s + (0.392 − 0.919i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3965406825 - 0.2077846505i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3965406825 - 0.2077846505i\) |
\(L(1)\) |
\(\approx\) |
\(0.6731607613 + 0.2118975594i\) |
\(L(1)\) |
\(\approx\) |
\(0.6731607613 + 0.2118975594i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (0.299 + 0.954i)T \) |
| 3 | \( 1 + (-0.556 - 0.831i)T \) |
| 5 | \( 1 + (0.503 + 0.863i)T \) |
| 7 | \( 1 + (-0.860 + 0.508i)T \) |
| 11 | \( 1 + (-0.972 - 0.233i)T \) |
| 13 | \( 1 + (-0.966 - 0.257i)T \) |
| 17 | \( 1 + (0.392 - 0.919i)T \) |
| 19 | \( 1 + (0.524 - 0.851i)T \) |
| 29 | \( 1 + (0.813 - 0.581i)T \) |
| 31 | \( 1 + (-0.759 + 0.650i)T \) |
| 37 | \( 1 + (0.879 - 0.476i)T \) |
| 41 | \( 1 + (0.645 - 0.763i)T \) |
| 43 | \( 1 + (-0.993 + 0.111i)T \) |
| 47 | \( 1 + (-0.990 - 0.136i)T \) |
| 53 | \( 1 + (-0.673 - 0.739i)T \) |
| 59 | \( 1 + (0.767 - 0.640i)T \) |
| 61 | \( 1 + (-0.0434 + 0.999i)T \) |
| 67 | \( 1 + (0.369 - 0.929i)T \) |
| 71 | \( 1 + (-0.834 - 0.551i)T \) |
| 73 | \( 1 + (0.813 + 0.581i)T \) |
| 79 | \( 1 + (0.566 - 0.824i)T \) |
| 83 | \( 1 + (-0.982 - 0.185i)T \) |
| 89 | \( 1 + (-0.358 + 0.933i)T \) |
| 97 | \( 1 + (0.154 - 0.987i)T \) |
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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
−23.462930815981686208108816725695, −22.58973776038552913126952196003, −21.74149900673317468906861740642, −21.233623138605670625705469705496, −20.309232169902706998544077072660, −19.84170548162836165431749909211, −18.62041450615653912004417866709, −17.65229424137986338237649568956, −16.810780649111932866914299347898, −16.176969682171170820214774574695, −14.99594581201971500318987627847, −14.097183321966900758950524526738, −12.899050022833179818794197424086, −12.57564138950157269842959956915, −11.54000978482213760671111865486, −10.3059530475165110267151463725, −9.97631016278843739056549395840, −9.27964479396773482808843035820, −8.031191672848352601526139146090, −6.30139088214924938149119062348, −5.430775131865489816667746683113, −4.68627263979998743430752407124, −3.77782949244694775262648643216, −2.68978610822609034104112206794, −1.226010303870498048236561471942,
0.24528245180916536285887338578, 2.50630952926975736467093337681, 3.09485123322878227436743376433, 5.05368647047468288676498483543, 5.57963129450564348853473117613, 6.57207322228981357223292719738, 7.174518267388174209195115362402, 7.9842989918540613005138335672, 9.349085440534339831911907479635, 10.16786720373154399216544227694, 11.451510001195394448418523987953, 12.443262215242581608771035969261, 13.188243304135843851335991563785, 13.85933331373115305093094684504, 14.794911413452809994743528194892, 15.828453846670429472435504119152, 16.467728295648059439028118186229, 17.633424033198151777299364155, 18.08296813321708683748078363763, 18.79800644826484725530060453826, 19.64016083340392857300184942967, 21.33299286787246730646923884912, 22.05479539046211548714199944872, 22.66001602557373366057574347212, 23.28958279953011370131373942039