| L(s) = 1 | − 2.63·2-s − 3-s + 4.92·4-s + 0.271·5-s + 2.63·6-s − 7-s − 7.70·8-s + 9-s − 0.714·10-s − 3.79·11-s − 4.92·12-s − 4.36·13-s + 2.63·14-s − 0.271·15-s + 10.4·16-s + 7.69·17-s − 2.63·18-s − 6.00·19-s + 1.33·20-s + 21-s + 9.98·22-s − 8.93·23-s + 7.70·24-s − 4.92·25-s + 11.4·26-s − 27-s − 4.92·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 0.577·3-s + 2.46·4-s + 0.121·5-s + 1.07·6-s − 0.377·7-s − 2.72·8-s + 0.333·9-s − 0.225·10-s − 1.14·11-s − 1.42·12-s − 1.21·13-s + 0.703·14-s − 0.0700·15-s + 2.60·16-s + 1.86·17-s − 0.620·18-s − 1.37·19-s + 0.298·20-s + 0.218·21-s + 2.12·22-s − 1.86·23-s + 1.57·24-s − 0.985·25-s + 2.25·26-s − 0.192·27-s − 0.931·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08816892581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08816892581\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 5 | \( 1 - 0.271T + 5T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 + 4.36T + 13T^{2} \) |
| 17 | \( 1 - 7.69T + 17T^{2} \) |
| 19 | \( 1 + 6.00T + 19T^{2} \) |
| 23 | \( 1 + 8.93T + 23T^{2} \) |
| 29 | \( 1 + 4.84T + 29T^{2} \) |
| 31 | \( 1 - 8.67T + 31T^{2} \) |
| 37 | \( 1 - 1.18T + 37T^{2} \) |
| 41 | \( 1 + 7.69T + 41T^{2} \) |
| 43 | \( 1 - 2.97T + 43T^{2} \) |
| 47 | \( 1 + 7.26T + 47T^{2} \) |
| 53 | \( 1 - 1.58T + 53T^{2} \) |
| 59 | \( 1 + 4.77T + 59T^{2} \) |
| 61 | \( 1 + 4.69T + 61T^{2} \) |
| 67 | \( 1 - 4.09T + 67T^{2} \) |
| 71 | \( 1 + 9.80T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 5.31T + 79T^{2} \) |
| 83 | \( 1 - 4.30T + 83T^{2} \) |
| 89 | \( 1 + 18.2T + 89T^{2} \) |
| 97 | \( 1 - 2.83T + 97T^{2} \) |
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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
−7.958100864476122954489209364194, −7.45167526148954663178647930540, −6.62363092689124224630606925298, −5.96294139537418254936062288043, −5.43085790434490470306637811728, −4.28882878825878009160471978682, −3.08122518721524307383874263977, −2.30686376328553241027008746416, −1.55530742743304030355358555560, −0.19587819004086724657546864342,
0.19587819004086724657546864342, 1.55530742743304030355358555560, 2.30686376328553241027008746416, 3.08122518721524307383874263977, 4.28882878825878009160471978682, 5.43085790434490470306637811728, 5.96294139537418254936062288043, 6.62363092689124224630606925298, 7.45167526148954663178647930540, 7.958100864476122954489209364194
