L(s) = 1 | + 2.70·2-s − 0.345·3-s + 5.30·4-s + 3.25·5-s − 0.935·6-s − 7-s + 8.94·8-s − 2.88·9-s + 8.80·10-s − 1.84·11-s − 1.83·12-s − 2.70·14-s − 1.12·15-s + 13.5·16-s + 2.15·17-s − 7.78·18-s − 2.40·19-s + 17.2·20-s + 0.345·21-s − 4.99·22-s + 1.81·23-s − 3.09·24-s + 5.61·25-s + 2.03·27-s − 5.30·28-s − 2.73·29-s − 3.04·30-s + ⋯ |
L(s) = 1 | + 1.91·2-s − 0.199·3-s + 2.65·4-s + 1.45·5-s − 0.381·6-s − 0.377·7-s + 3.16·8-s − 0.960·9-s + 2.78·10-s − 0.556·11-s − 0.530·12-s − 0.722·14-s − 0.291·15-s + 3.38·16-s + 0.522·17-s − 1.83·18-s − 0.550·19-s + 3.86·20-s + 0.0754·21-s − 1.06·22-s + 0.377·23-s − 0.631·24-s + 1.12·25-s + 0.391·27-s − 1.00·28-s − 0.507·29-s − 0.556·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.688380515\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.688380515\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 3 | \( 1 + 0.345T + 3T^{2} \) |
| 5 | \( 1 - 3.25T + 5T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 19 | \( 1 + 2.40T + 19T^{2} \) |
| 23 | \( 1 - 1.81T + 23T^{2} \) |
| 29 | \( 1 + 2.73T + 29T^{2} \) |
| 31 | \( 1 - 1.74T + 31T^{2} \) |
| 37 | \( 1 + 5.93T + 37T^{2} \) |
| 41 | \( 1 + 4.22T + 41T^{2} \) |
| 43 | \( 1 + 8.68T + 43T^{2} \) |
| 47 | \( 1 - 5.87T + 47T^{2} \) |
| 53 | \( 1 + 9.30T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 0.826T + 67T^{2} \) |
| 71 | \( 1 + 2.35T + 71T^{2} \) |
| 73 | \( 1 - 3.19T + 73T^{2} \) |
| 79 | \( 1 - 0.801T + 79T^{2} \) |
| 83 | \( 1 - 9.97T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 9.23T + 97T^{2} \) |
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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
−10.18563410189934767763023775303, −9.040631196154107389098727669982, −7.81554528285669634769456335051, −6.63234187601328730482727216846, −6.18855320772094710507074814288, −5.38214148681914448056464508528, −4.97198756547918707475596748080, −3.52586849597655241413870142372, −2.73130567511854820079895814731, −1.84079636640731647897466443701,
1.84079636640731647897466443701, 2.73130567511854820079895814731, 3.52586849597655241413870142372, 4.97198756547918707475596748080, 5.38214148681914448056464508528, 6.18855320772094710507074814288, 6.63234187601328730482727216846, 7.81554528285669634769456335051, 9.040631196154107389098727669982, 10.18563410189934767763023775303