| L(s) = 1 | + 2-s − 2.23·3-s + 4-s + 5-s − 2.23·6-s − 0.971·7-s + 8-s + 1.99·9-s + 10-s − 2.26·11-s − 2.23·12-s + 4.74·13-s − 0.971·14-s − 2.23·15-s + 16-s − 5.16·17-s + 1.99·18-s − 2.10·19-s + 20-s + 2.17·21-s − 2.26·22-s + 3.32·23-s − 2.23·24-s + 25-s + 4.74·26-s + 2.24·27-s − 0.971·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.29·3-s + 0.5·4-s + 0.447·5-s − 0.912·6-s − 0.367·7-s + 0.353·8-s + 0.664·9-s + 0.316·10-s − 0.684·11-s − 0.645·12-s + 1.31·13-s − 0.259·14-s − 0.577·15-s + 0.250·16-s − 1.25·17-s + 0.470·18-s − 0.482·19-s + 0.223·20-s + 0.474·21-s − 0.483·22-s + 0.693·23-s − 0.456·24-s + 0.200·25-s + 0.931·26-s + 0.432·27-s − 0.183·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 7 | \( 1 + 0.971T + 7T^{2} \) |
| 11 | \( 1 + 2.26T + 11T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 17 | \( 1 + 5.16T + 17T^{2} \) |
| 19 | \( 1 + 2.10T + 19T^{2} \) |
| 23 | \( 1 - 3.32T + 23T^{2} \) |
| 29 | \( 1 + 7.06T + 29T^{2} \) |
| 31 | \( 1 - 5.27T + 31T^{2} \) |
| 37 | \( 1 + 6.88T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 5.15T + 43T^{2} \) |
| 47 | \( 1 + 2.01T + 47T^{2} \) |
| 53 | \( 1 + 9.34T + 53T^{2} \) |
| 59 | \( 1 - 8.51T + 59T^{2} \) |
| 61 | \( 1 + 3.05T + 61T^{2} \) |
| 67 | \( 1 + 4.20T + 67T^{2} \) |
| 71 | \( 1 + 3.81T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 2.24T + 89T^{2} \) |
| 97 | \( 1 + 8.88T + 97T^{2} \) |
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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
−7.39184696072079367377925165119, −6.69665716263825871679888343351, −6.03661394810270604147196214193, −5.80063138172291776044703895429, −4.87867383424718891826193881659, −4.32316240086379864259877887290, −3.31000262137355842980454493013, −2.39113271055716841462225442808, −1.29340191557793768935033068759, 0,
1.29340191557793768935033068759, 2.39113271055716841462225442808, 3.31000262137355842980454493013, 4.32316240086379864259877887290, 4.87867383424718891826193881659, 5.80063138172291776044703895429, 6.03661394810270604147196214193, 6.69665716263825871679888343351, 7.39184696072079367377925165119
