| L(s) = 1 | + 0.571·2-s − 1.67·4-s − 2.42·7-s − 2.10·8-s − 1.14·11-s − 1.57·13-s − 1.38·14-s + 2.14·16-s + 4.67·17-s + 5.34·19-s − 0.654·22-s − 1.81·23-s − 0.899·26-s + 4.06·28-s − 1.95·29-s + 31-s + 5.42·32-s + 2.67·34-s − 3.57·37-s + 3.05·38-s + 3.14·41-s − 2.85·43-s + 1.91·44-s − 1.03·46-s + 7.16·47-s − 1.10·49-s + 2.62·52-s + ⋯ |
| L(s) = 1 | + 0.404·2-s − 0.836·4-s − 0.917·7-s − 0.742·8-s − 0.344·11-s − 0.435·13-s − 0.371·14-s + 0.535·16-s + 1.13·17-s + 1.22·19-s − 0.139·22-s − 0.378·23-s − 0.176·26-s + 0.767·28-s − 0.363·29-s + 0.179·31-s + 0.959·32-s + 0.458·34-s − 0.587·37-s + 0.496·38-s + 0.491·41-s − 0.435·43-s + 0.288·44-s − 0.153·46-s + 1.04·47-s − 0.157·49-s + 0.364·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.571T + 2T^{2} \) |
| 7 | \( 1 + 2.42T + 7T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 17 | \( 1 - 4.67T + 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 + 1.81T + 23T^{2} \) |
| 29 | \( 1 + 1.95T + 29T^{2} \) |
| 37 | \( 1 + 3.57T + 37T^{2} \) |
| 41 | \( 1 - 3.14T + 41T^{2} \) |
| 43 | \( 1 + 2.85T + 43T^{2} \) |
| 47 | \( 1 - 7.16T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 0.917T + 67T^{2} \) |
| 71 | \( 1 - 6.44T + 71T^{2} \) |
| 73 | \( 1 + 6.71T + 73T^{2} \) |
| 79 | \( 1 - 9.52T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−7.66195903231686801864534179366, −6.87428527845573863017337086635, −5.98554766316627719078278177930, −5.43320902250591339258467182789, −4.85792984740046202021297321787, −3.83123315366894664058616670791, −3.34590824963253091979956212094, −2.57121363586191369805889463996, −1.10527275274349620774310162147, 0,
1.10527275274349620774310162147, 2.57121363586191369805889463996, 3.34590824963253091979956212094, 3.83123315366894664058616670791, 4.85792984740046202021297321787, 5.43320902250591339258467182789, 5.98554766316627719078278177930, 6.87428527845573863017337086635, 7.66195903231686801864534179366
