| L(s) = 1 | + 3i·3-s + (−2 − 11i)5-s + 2i·7-s − 9·9-s + 70·11-s + 54i·13-s + (33 − 6i)15-s − 22i·17-s − 24·19-s − 6·21-s − 100i·23-s + (−117 + 44i)25-s − 27i·27-s + 216·29-s − 208·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.178 − 0.983i)5-s + 0.107i·7-s − 0.333·9-s + 1.91·11-s + 1.15i·13-s + (0.568 − 0.103i)15-s − 0.313i·17-s − 0.289·19-s − 0.0623·21-s − 0.906i·23-s + (−0.936 + 0.351i)25-s − 0.192i·27-s + 1.38·29-s − 1.20·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.178599532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.178599532\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 + (2 + 11i)T \) |
| good | 7 | \( 1 - 2iT - 343T^{2} \) |
| 11 | \( 1 - 70T + 1.33e3T^{2} \) |
| 13 | \( 1 - 54iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 22iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 24T + 6.85e3T^{2} \) |
| 23 | \( 1 + 100iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 216T + 2.43e4T^{2} \) |
| 31 | \( 1 + 208T + 2.97e4T^{2} \) |
| 37 | \( 1 - 254iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 206T + 6.89e4T^{2} \) |
| 43 | \( 1 + 292iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 320iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 402iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 370T + 2.05e5T^{2} \) |
| 61 | \( 1 - 550T + 2.26e5T^{2} \) |
| 67 | \( 1 - 728iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 540T + 3.57e5T^{2} \) |
| 73 | \( 1 + 604iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 792T + 4.93e5T^{2} \) |
| 83 | \( 1 + 404iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 938T + 7.04e5T^{2} \) |
| 97 | \( 1 - 56iT - 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.443027152839833840281802098872, −8.916077389744112995404585936578, −8.382032193974636905220141939523, −6.95119104737054049554836208855, −6.30600560368817961417891502801, −5.07533428766961604031997568931, −4.31465635229975468909707521738, −3.65898375533114537101263828948, −1.99198318130975708246336157194, −0.828121072329971548702486123571,
0.808528621866258996298362676715, 2.02981178325438252640846382049, 3.29874965429420689782219854862, 3.98150145691963405328867845134, 5.51420709992944018050773008002, 6.39808282576933830047452847566, 6.99164220618287742172170881065, 7.81333637005420609414859209230, 8.717425008929317773583590780597, 9.640015039089004751355671340319
