| L(s) = 1 | − 2.82·2-s − 3-s + 5.95·4-s − 4.23·5-s + 2.82·6-s − 7-s − 11.1·8-s + 9-s + 11.9·10-s − 3.53·11-s − 5.95·12-s + 5.69·13-s + 2.82·14-s + 4.23·15-s + 19.5·16-s + 6.11·17-s − 2.82·18-s − 0.959·19-s − 25.2·20-s + 21-s + 9.96·22-s + 0.804·23-s + 11.1·24-s + 12.9·25-s − 16.0·26-s − 27-s − 5.95·28-s + ⋯ |
| L(s) = 1 | − 1.99·2-s − 0.577·3-s + 2.97·4-s − 1.89·5-s + 1.15·6-s − 0.377·7-s − 3.94·8-s + 0.333·9-s + 3.77·10-s − 1.06·11-s − 1.72·12-s + 1.57·13-s + 0.753·14-s + 1.09·15-s + 4.89·16-s + 1.48·17-s − 0.664·18-s − 0.220·19-s − 5.64·20-s + 0.218·21-s + 2.12·22-s + 0.167·23-s + 2.27·24-s + 2.58·25-s − 3.15·26-s − 0.192·27-s − 1.12·28-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
| \(L(1)\) |
\(\approx\) |
\(0.4899675234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4899675234\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.82T + 2T^{2} \) |
| 5 | \( 1 + 4.23T + 5T^{2} \) |
| 11 | \( 1 + 3.53T + 11T^{2} \) |
| 13 | \( 1 - 5.69T + 13T^{2} \) |
| 17 | \( 1 - 6.11T + 17T^{2} \) |
| 19 | \( 1 + 0.959T + 19T^{2} \) |
| 23 | \( 1 - 0.804T + 23T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 - 0.289T + 31T^{2} \) |
| 37 | \( 1 - 4.28T + 37T^{2} \) |
| 41 | \( 1 - 2.94T + 41T^{2} \) |
| 43 | \( 1 + 1.74T + 43T^{2} \) |
| 47 | \( 1 - 8.28T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 8.02T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 0.616T + 71T^{2} \) |
| 73 | \( 1 + 9.19T + 73T^{2} \) |
| 79 | \( 1 + 5.58T + 79T^{2} \) |
| 83 | \( 1 - 0.412T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 2.08T + 97T^{2} \) |
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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
−7.977550462385840656818679173622, −7.37321709237070441927421637785, −6.85357183758431981288955122641, −6.06463928675731809696623453971, −5.35027001138960756804687105352, −3.96200698495557984710400944463, −3.32084283827587941217776933442, −2.53992890464687056333911191055, −0.939886873193587218747068233735, −0.68469917044945457590894591164,
0.68469917044945457590894591164, 0.939886873193587218747068233735, 2.53992890464687056333911191055, 3.32084283827587941217776933442, 3.96200698495557984710400944463, 5.35027001138960756804687105352, 6.06463928675731809696623453971, 6.85357183758431981288955122641, 7.37321709237070441927421637785, 7.977550462385840656818679173622
