| L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s − 2·7-s − 8-s + 9-s + 3·10-s − 11-s − 12-s − 4·13-s + 2·14-s + 3·15-s + 16-s + 3·17-s − 18-s + 4·19-s − 3·20-s + 2·21-s + 22-s − 4·23-s + 24-s + 4·25-s + 4·26-s − 27-s − 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.301·11-s − 0.288·12-s − 1.10·13-s + 0.534·14-s + 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.917·19-s − 0.670·20-s + 0.436·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s + 4/5·25-s + 0.784·26-s − 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4773519098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4773519098\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.92134100606093, −13.17066752672349, −12.67817058212864, −12.17988006425799, −11.78632301971450, −11.59780804723059, −10.91271744107605, −10.23258733281469, −9.845005571560701, −9.704383867231517, −8.745033995199024, −8.244710970847152, −7.789369314851031, −7.356097314918725, −6.858765096664944, −6.399320754662564, −5.633029645422425, −5.098416216580242, −4.545813086151272, −3.768474193652139, −3.297928027161587, −2.724176208151212, −1.884991477076292, −0.8869773553054354, −0.3362826390118360,
0.3362826390118360, 0.8869773553054354, 1.884991477076292, 2.724176208151212, 3.297928027161587, 3.768474193652139, 4.545813086151272, 5.098416216580242, 5.633029645422425, 6.399320754662564, 6.858765096664944, 7.356097314918725, 7.789369314851031, 8.244710970847152, 8.745033995199024, 9.704383867231517, 9.845005571560701, 10.23258733281469, 10.91271744107605, 11.59780804723059, 11.78632301971450, 12.17988006425799, 12.67817058212864, 13.17066752672349, 13.92134100606093
