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 | \( 1 + 56iT - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.640015039089004751355671340319, −8.717425008929317773583590780597, −7.81333637005420609414859209230, −6.99164220618287742172170881065, −6.39808282576933830047452847566, −5.51420709992944018050773008002, −3.98150145691963405328867845134, −3.29874965429420689782219854862, −2.02981178325438252640846382049, −0.808528621866258996298362676715,
0.828121072329971548702486123571, 1.99198318130975708246336157194, 3.65898375533114537101263828948, 4.31465635229975468909707521738, 5.07533428766961604031997568931, 6.30600560368817961417891502801, 6.95119104737054049554836208855, 8.382032193974636905220141939523, 8.916077389744112995404585936578, 9.443027152839833840281802098872