L(s) = 1 | + 2-s − 3-s + 4-s + 4.36·5-s − 6-s − 4.18·7-s + 8-s + 9-s + 4.36·10-s − 0.462·11-s − 12-s − 5.66·13-s − 4.18·14-s − 4.36·15-s + 16-s − 7.64·17-s + 18-s + 19-s + 4.36·20-s + 4.18·21-s − 0.462·22-s + 5.05·23-s − 24-s + 14.0·25-s − 5.66·26-s − 27-s − 4.18·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.95·5-s − 0.408·6-s − 1.58·7-s + 0.353·8-s + 0.333·9-s + 1.38·10-s − 0.139·11-s − 0.288·12-s − 1.57·13-s − 1.11·14-s − 1.12·15-s + 0.250·16-s − 1.85·17-s + 0.235·18-s + 0.229·19-s + 0.977·20-s + 0.912·21-s − 0.0986·22-s + 1.05·23-s − 0.204·24-s + 2.81·25-s − 1.11·26-s − 0.192·27-s − 0.790·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
good | 5 | \( 1 - 4.36T + 5T^{2} \) |
| 7 | \( 1 + 4.18T + 7T^{2} \) |
| 11 | \( 1 + 0.462T + 11T^{2} \) |
| 13 | \( 1 + 5.66T + 13T^{2} \) |
| 17 | \( 1 + 7.64T + 17T^{2} \) |
| 23 | \( 1 - 5.05T + 23T^{2} \) |
| 29 | \( 1 - 0.972T + 29T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 37 | \( 1 + 8.67T + 37T^{2} \) |
| 41 | \( 1 + 3.87T + 41T^{2} \) |
| 43 | \( 1 + 7.90T + 43T^{2} \) |
| 47 | \( 1 - 5.36T + 47T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 2.69T + 61T^{2} \) |
| 67 | \( 1 + 8.85T + 67T^{2} \) |
| 71 | \( 1 + 9.10T + 71T^{2} \) |
| 73 | \( 1 + 8.12T + 73T^{2} \) |
| 79 | \( 1 + 3.22T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.06862134469458689561946877082, −6.76014940263025623917957078423, −6.39334340546310831676622505773, −5.45402476304217974115540980974, −5.11927225856459254566322215102, −4.26323015708436815768576620431, −2.83171110759526534127373046526, −2.64324077324142926096735952585, −1.56201830513640398649758287821, 0,
1.56201830513640398649758287821, 2.64324077324142926096735952585, 2.83171110759526534127373046526, 4.26323015708436815768576620431, 5.11927225856459254566322215102, 5.45402476304217974115540980974, 6.39334340546310831676622505773, 6.76014940263025623917957078423, 7.06862134469458689561946877082