| L(s) = 1 | − 2-s − 1.11·3-s + 4-s + 4.02·5-s + 1.11·6-s + 1.02·7-s − 8-s − 1.75·9-s − 4.02·10-s + 0.681·11-s − 1.11·12-s + 3.81·13-s − 1.02·14-s − 4.48·15-s + 16-s − 3.05·17-s + 1.75·18-s + 19-s + 4.02·20-s − 1.13·21-s − 0.681·22-s + 3.47·23-s + 1.11·24-s + 11.2·25-s − 3.81·26-s + 5.30·27-s + 1.02·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.643·3-s + 0.5·4-s + 1.80·5-s + 0.454·6-s + 0.386·7-s − 0.353·8-s − 0.586·9-s − 1.27·10-s + 0.205·11-s − 0.321·12-s + 1.05·13-s − 0.272·14-s − 1.15·15-s + 0.250·16-s − 0.741·17-s + 0.414·18-s + 0.229·19-s + 0.900·20-s − 0.248·21-s − 0.145·22-s + 0.725·23-s + 0.227·24-s + 2.24·25-s − 0.747·26-s + 1.02·27-s + 0.193·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.976447620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.976447620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 1.11T + 3T^{2} \) |
| 5 | \( 1 - 4.02T + 5T^{2} \) |
| 7 | \( 1 - 1.02T + 7T^{2} \) |
| 11 | \( 1 - 0.681T + 11T^{2} \) |
| 13 | \( 1 - 3.81T + 13T^{2} \) |
| 17 | \( 1 + 3.05T + 17T^{2} \) |
| 23 | \( 1 - 3.47T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 9.61T + 31T^{2} \) |
| 37 | \( 1 + 2.81T + 37T^{2} \) |
| 41 | \( 1 + 6.96T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 4.96T + 47T^{2} \) |
| 53 | \( 1 + 1.46T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 + 0.225T + 61T^{2} \) |
| 67 | \( 1 + 6.68T + 67T^{2} \) |
| 71 | \( 1 + 1.07T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 1.21T + 79T^{2} \) |
| 83 | \( 1 - 5.91T + 83T^{2} \) |
| 89 | \( 1 - 2.83T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.087425187444649950156782291959, −6.77942683445181710534835220322, −6.42164118660997936677798396583, −6.01820507129075051370428735531, −5.15556312897949907711801695261, −4.65377079693052068292298094480, −3.14066836387920888190403080062, −2.50929049814642775181162890125, −1.52700596723752820724927673393, −0.874092593339416250534101745063,
0.874092593339416250534101745063, 1.52700596723752820724927673393, 2.50929049814642775181162890125, 3.14066836387920888190403080062, 4.65377079693052068292298094480, 5.15556312897949907711801695261, 6.01820507129075051370428735531, 6.42164118660997936677798396583, 6.77942683445181710534835220322, 8.087425187444649950156782291959
