L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)4-s + (0.608 − 0.793i)5-s + (0.608 + 0.793i)7-s + (−0.707 + 0.707i)8-s + (−0.382 + 0.923i)10-s + (−0.793 + 0.608i)11-s + (−0.866 + 0.5i)13-s + (−0.793 − 0.608i)14-s + (0.5 − 0.866i)16-s + (0.707 + 0.707i)19-s + (0.130 − 0.991i)20-s + (0.608 − 0.793i)22-s + (0.130 + 0.991i)23-s + (−0.258 − 0.965i)25-s + (0.707 − 0.707i)26-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)4-s + (0.608 − 0.793i)5-s + (0.608 + 0.793i)7-s + (−0.707 + 0.707i)8-s + (−0.382 + 0.923i)10-s + (−0.793 + 0.608i)11-s + (−0.866 + 0.5i)13-s + (−0.793 − 0.608i)14-s + (0.5 − 0.866i)16-s + (0.707 + 0.707i)19-s + (0.130 − 0.991i)20-s + (0.608 − 0.793i)22-s + (0.130 + 0.991i)23-s + (−0.258 − 0.965i)25-s + (0.707 − 0.707i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0524 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0524 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6949259062 + 0.7323582460i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6949259062 + 0.7323582460i\) |
\(L(1)\) |
\(\approx\) |
\(0.7422110670 + 0.1938113578i\) |
\(L(1)\) |
\(\approx\) |
\(0.7422110670 + 0.1938113578i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 5 | \( 1 + (0.608 - 0.793i)T \) |
| 7 | \( 1 + (0.608 + 0.793i)T \) |
| 11 | \( 1 + (-0.793 + 0.608i)T \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
| 19 | \( 1 + (0.707 + 0.707i)T \) |
| 23 | \( 1 + (0.130 + 0.991i)T \) |
| 29 | \( 1 + (0.991 + 0.130i)T \) |
| 31 | \( 1 + (-0.793 - 0.608i)T \) |
| 37 | \( 1 + (-0.923 + 0.382i)T \) |
| 41 | \( 1 + (-0.991 + 0.130i)T \) |
| 43 | \( 1 + (0.258 + 0.965i)T \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.707 + 0.707i)T \) |
| 59 | \( 1 + (0.965 + 0.258i)T \) |
| 61 | \( 1 + (-0.608 - 0.793i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.923 + 0.382i)T \) |
| 73 | \( 1 + (0.382 + 0.923i)T \) |
| 79 | \( 1 + (-0.793 + 0.608i)T \) |
| 83 | \( 1 + (0.965 - 0.258i)T \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 + (0.991 + 0.130i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.11530614119596609216690866802, −26.76773349997378935094656739981, −25.85805017213841555553061935762, −24.77156106023739981206128522648, −23.83887627613601323462483805712, −22.358840718495103832524248417000, −21.412120200089921518487209423142, −20.51131504002601036414531701893, −19.50393182878101711040680643326, −18.3742216880118520429804811970, −17.707020051622149963394043514156, −16.83240854685135516420869268758, −15.59275365636991657617572222535, −14.41028327956108567351368740678, −13.3076089358593767341215067504, −11.86053403514742741602846422464, −10.552553616468519815215674149460, −10.35504934078243966441342187105, −8.8350233412979806888583768235, −7.60529654106459334906374806219, −6.82125200965514550864296512271, −5.25785867149256300811626415144, −3.27369303322880075436292547753, −2.182799420257388439000187144706, −0.51970311614797170661323337079,
1.44578273745255881547722847872, 2.45978794835728570868889769966, 4.95512553522715836706196413974, 5.73840924800919573005796862549, 7.31220662909244338183974544555, 8.33423383369965168531770134339, 9.36786910172090814553796132244, 10.14090770899420828902911336204, 11.64049927369356018693832133698, 12.49109254710517847696341891621, 14.05948656475005959775461506791, 15.186302392426864739741050645910, 16.13199313288179487625794255489, 17.22582798399488225675934539728, 17.930009246460432997709023083, 18.86866718659772369975403679125, 20.126752677089128070734956916, 20.92882826380477124108285840652, 21.800536836904004369816467608391, 23.550947361901781387603095547287, 24.42170962937320494463896976039, 25.12796259033887379910709895938, 25.97502798255220964781734006474, 27.18583109947936505270069972028, 27.97026765236815705540338212780