L(s) = 1 | + 0.583·2-s − 3-s − 1.65·4-s − 0.583·6-s + 4.38·7-s − 2.13·8-s + 9-s + 1.65·12-s − 1.72·13-s + 2.56·14-s + 2.07·16-s + 3.73·17-s + 0.583·18-s + 2.79·19-s − 4.38·21-s + 6.45·23-s + 2.13·24-s − 1.00·26-s − 27-s − 7.28·28-s + 7.88·29-s + 3.05·31-s + 5.48·32-s + 2.17·34-s − 1.65·36-s − 9.21·37-s + 1.62·38-s + ⋯ |
L(s) = 1 | + 0.412·2-s − 0.577·3-s − 0.829·4-s − 0.238·6-s + 1.65·7-s − 0.755·8-s + 0.333·9-s + 0.478·12-s − 0.477·13-s + 0.684·14-s + 0.517·16-s + 0.904·17-s + 0.137·18-s + 0.640·19-s − 0.957·21-s + 1.34·23-s + 0.436·24-s − 0.196·26-s − 0.192·27-s − 1.37·28-s + 1.46·29-s + 0.548·31-s + 0.969·32-s + 0.373·34-s − 0.276·36-s − 1.51·37-s + 0.264·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.229934039\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.229934039\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.583T + 2T^{2} \) |
| 7 | \( 1 - 4.38T + 7T^{2} \) |
| 13 | \( 1 + 1.72T + 13T^{2} \) |
| 17 | \( 1 - 3.73T + 17T^{2} \) |
| 19 | \( 1 - 2.79T + 19T^{2} \) |
| 23 | \( 1 - 6.45T + 23T^{2} \) |
| 29 | \( 1 - 7.88T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 + 9.20T + 41T^{2} \) |
| 43 | \( 1 - 8.88T + 43T^{2} \) |
| 47 | \( 1 - 6.29T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 + 0.0655T + 59T^{2} \) |
| 61 | \( 1 + 5.21T + 61T^{2} \) |
| 67 | \( 1 - 6.67T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 - 3.40T + 73T^{2} \) |
| 79 | \( 1 + 3.56T + 79T^{2} \) |
| 83 | \( 1 - 7.22T + 83T^{2} \) |
| 89 | \( 1 - 9.20T + 89T^{2} \) |
| 97 | \( 1 + 3.33T + 97T^{2} \) |
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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
−7.68329561926526674634415678463, −7.14367767618500457361535965875, −6.14570459496464368945855928701, −5.36353753903439478403777405419, −4.92074678264504966920821863045, −4.60677991127970847949027720105, −3.60291051095973213102465040740, −2.76565405831674480733394497716, −1.49058985337220050969917731063, −0.78028295045279131063153540566,
0.78028295045279131063153540566, 1.49058985337220050969917731063, 2.76565405831674480733394497716, 3.60291051095973213102465040740, 4.60677991127970847949027720105, 4.92074678264504966920821863045, 5.36353753903439478403777405419, 6.14570459496464368945855928701, 7.14367767618500457361535965875, 7.68329561926526674634415678463