| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s + 4·11-s − 2·13-s + 4·14-s + 16-s − 2·17-s − 4·22-s − 8·23-s + 2·26-s − 4·28-s + 6·29-s − 4·31-s − 32-s + 2·34-s − 10·37-s − 2·41-s − 12·43-s + 4·44-s + 8·46-s + 9·49-s − 2·52-s − 6·53-s + 4·56-s − 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s + 1.20·11-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s − 0.852·22-s − 1.66·23-s + 0.392·26-s − 0.755·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.342·34-s − 1.64·37-s − 0.312·41-s − 1.82·43-s + 0.603·44-s + 1.17·46-s + 9/7·49-s − 0.277·52-s − 0.824·53-s + 0.534·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.51871091868931, −12.96384800828386, −12.34485873863523, −12.07251163666544, −11.74547586058191, −11.09513511439570, −10.32359103905886, −10.14872933700015, −9.746653945964186, −9.161039739108662, −8.806347720134269, −8.392893703832451, −7.587258744963104, −7.211656262893584, −6.583398038069218, −6.317719567791524, −5.974094108202790, −5.068374440295104, −4.519348107291053, −3.745510521192626, −3.381754887176007, −2.872264440755780, −1.925431481696613, −1.685973572035595, −0.5677055718849750, 0,
0.5677055718849750, 1.685973572035595, 1.925431481696613, 2.872264440755780, 3.381754887176007, 3.745510521192626, 4.519348107291053, 5.068374440295104, 5.974094108202790, 6.317719567791524, 6.583398038069218, 7.211656262893584, 7.587258744963104, 8.392893703832451, 8.806347720134269, 9.161039739108662, 9.746653945964186, 10.14872933700015, 10.32359103905886, 11.09513511439570, 11.74547586058191, 12.07251163666544, 12.34485873863523, 12.96384800828386, 13.51871091868931
