L(s) = 1 | − 3.10·3-s − 1.10·5-s + 7-s + 6.62·9-s + 5.62·11-s − 3.62·13-s + 3.42·15-s + 4.52·17-s + 0.578·19-s − 3.10·21-s + 23-s − 3.78·25-s − 11.2·27-s + 5.83·29-s + 2.52·31-s − 17.4·33-s − 1.10·35-s − 7.04·37-s + 11.2·39-s + 3.15·41-s − 7.25·43-s − 7.30·45-s − 2.52·47-s + 49-s − 14.0·51-s + 3.04·53-s − 6.20·55-s + ⋯ |
L(s) = 1 | − 1.79·3-s − 0.493·5-s + 0.377·7-s + 2.20·9-s + 1.69·11-s − 1.00·13-s + 0.883·15-s + 1.09·17-s + 0.132·19-s − 0.677·21-s + 0.208·23-s − 0.756·25-s − 2.16·27-s + 1.08·29-s + 0.453·31-s − 3.03·33-s − 0.186·35-s − 1.15·37-s + 1.80·39-s + 0.492·41-s − 1.10·43-s − 1.08·45-s − 0.368·47-s + 0.142·49-s − 1.96·51-s + 0.418·53-s − 0.836·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9320470727\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9320470727\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 3.10T + 3T^{2} \) |
| 5 | \( 1 + 1.10T + 5T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 + 3.62T + 13T^{2} \) |
| 17 | \( 1 - 4.52T + 17T^{2} \) |
| 19 | \( 1 - 0.578T + 19T^{2} \) |
| 29 | \( 1 - 5.83T + 29T^{2} \) |
| 31 | \( 1 - 2.52T + 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 - 3.15T + 41T^{2} \) |
| 43 | \( 1 + 7.25T + 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 - 3.04T + 53T^{2} \) |
| 59 | \( 1 + 9.30T + 59T^{2} \) |
| 61 | \( 1 - 8.35T + 61T^{2} \) |
| 67 | \( 1 + 5.62T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 6.72T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−8.984504522808245870743735980453, −7.906537954169173821462869590803, −7.13495704099955629233522267198, −6.55334178949962138238913310654, −5.79366280502407351500523148595, −4.96705503958532455577333354640, −4.38551670945015902178073712930, −3.44913261494807181663507137648, −1.66958168007837879843329567486, −0.70748537752957380519399015232,
0.70748537752957380519399015232, 1.66958168007837879843329567486, 3.44913261494807181663507137648, 4.38551670945015902178073712930, 4.96705503958532455577333354640, 5.79366280502407351500523148595, 6.55334178949962138238913310654, 7.13495704099955629233522267198, 7.906537954169173821462869590803, 8.984504522808245870743735980453