L(s) = 1 | − 2.53·5-s − 7-s + 0.467·11-s + 5.82·13-s − 3.87·17-s + 2.18·19-s − 0.106·23-s + 1.41·25-s − 8.78·29-s + 7.68·31-s + 2.53·35-s − 7.68·37-s + 2.22·41-s − 1.22·43-s − 5.33·47-s + 49-s + 0.716·53-s − 1.18·55-s + 0.736·59-s + 0.958·61-s − 14.7·65-s + 9.63·67-s + 13.2·71-s − 10.2·73-s − 0.467·77-s + 12.6·79-s − 2.73·83-s + ⋯ |
L(s) = 1 | − 1.13·5-s − 0.377·7-s + 0.141·11-s + 1.61·13-s − 0.940·17-s + 0.501·19-s − 0.0221·23-s + 0.282·25-s − 1.63·29-s + 1.37·31-s + 0.428·35-s − 1.26·37-s + 0.347·41-s − 0.187·43-s − 0.777·47-s + 0.142·49-s + 0.0984·53-s − 0.159·55-s + 0.0958·59-s + 0.122·61-s − 1.82·65-s + 1.17·67-s + 1.57·71-s − 1.20·73-s − 0.0533·77-s + 1.42·79-s − 0.299·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 2.53T + 5T^{2} \) |
| 11 | \( 1 - 0.467T + 11T^{2} \) |
| 13 | \( 1 - 5.82T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 - 2.18T + 19T^{2} \) |
| 23 | \( 1 + 0.106T + 23T^{2} \) |
| 29 | \( 1 + 8.78T + 29T^{2} \) |
| 31 | \( 1 - 7.68T + 31T^{2} \) |
| 37 | \( 1 + 7.68T + 37T^{2} \) |
| 41 | \( 1 - 2.22T + 41T^{2} \) |
| 43 | \( 1 + 1.22T + 43T^{2} \) |
| 47 | \( 1 + 5.33T + 47T^{2} \) |
| 53 | \( 1 - 0.716T + 53T^{2} \) |
| 59 | \( 1 - 0.736T + 59T^{2} \) |
| 61 | \( 1 - 0.958T + 61T^{2} \) |
| 67 | \( 1 - 9.63T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 2.73T + 83T^{2} \) |
| 89 | \( 1 - 8.11T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−7.39461633405108169721374695461, −6.71499005643958701631963020505, −6.15580742549110452741072774481, −5.31190648056582993354416313286, −4.43675889862898862476441443736, −3.68490593594605929558762716827, −3.40819644287094852944877013445, −2.19580330199352117706180210153, −1.10062158697882348182059365399, 0,
1.10062158697882348182059365399, 2.19580330199352117706180210153, 3.40819644287094852944877013445, 3.68490593594605929558762716827, 4.43675889862898862476441443736, 5.31190648056582993354416313286, 6.15580742549110452741072774481, 6.71499005643958701631963020505, 7.39461633405108169721374695461