| L(s) = 1 | − 2-s − 2.23·3-s + 4-s − 2.23·5-s + 2.23·6-s − 3·7-s − 8-s + 2.00·9-s + 2.23·10-s + 1.23·11-s − 2.23·12-s + 2.47·13-s + 3·14-s + 5.00·15-s + 16-s − 17-s − 2.00·18-s − 6.23·19-s − 2.23·20-s + 6.70·21-s − 1.23·22-s + 8.47·23-s + 2.23·24-s − 2.47·26-s + 2.23·27-s − 3·28-s + 2.23·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.29·3-s + 0.5·4-s − 0.999·5-s + 0.912·6-s − 1.13·7-s − 0.353·8-s + 0.666·9-s + 0.707·10-s + 0.372·11-s − 0.645·12-s + 0.685·13-s + 0.801·14-s + 1.29·15-s + 0.250·16-s − 0.242·17-s − 0.471·18-s − 1.43·19-s − 0.499·20-s + 1.46·21-s − 0.263·22-s + 1.76·23-s + 0.456·24-s − 0.484·26-s + 0.430·27-s − 0.566·28-s + 0.415·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 - 2.47T + 13T^{2} \) |
| 19 | \( 1 + 6.23T + 19T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 29 | \( 1 - 2.23T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 9.70T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 + 2.52T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 6.76T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.763734025615692772218910654037, −8.068560226717213195167327750083, −6.90777994733186257677037951042, −6.56992187281109649941001658583, −5.87277916100669029951386385760, −4.70562439557233136883755649080, −3.81833883699967120490146280014, −2.77507388462698710971565647887, −0.990782892342428255822049910678, 0,
0.990782892342428255822049910678, 2.77507388462698710971565647887, 3.81833883699967120490146280014, 4.70562439557233136883755649080, 5.87277916100669029951386385760, 6.56992187281109649941001658583, 6.90777994733186257677037951042, 8.068560226717213195167327750083, 8.763734025615692772218910654037
