L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 4·11-s − 4·13-s + 14-s + 16-s + 4·17-s − 2·19-s + 4·22-s − 23-s − 4·26-s + 28-s + 6·29-s − 4·31-s + 32-s + 4·34-s − 2·37-s − 2·38-s + 10·41-s − 12·43-s + 4·44-s − 46-s + 49-s − 4·52-s − 4·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.20·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.458·19-s + 0.852·22-s − 0.208·23-s − 0.784·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.685·34-s − 0.328·37-s − 0.324·38-s + 1.56·41-s − 1.82·43-s + 0.603·44-s − 0.147·46-s + 1/7·49-s − 0.554·52-s − 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.44435914318853, −13.99417501021348, −13.36957535022359, −12.82297868138700, −12.28188888481851, −11.88250543910161, −11.64790120775689, −10.89061842742700, −10.37009794157873, −9.876216775220684, −9.348776181364855, −8.746481668302728, −8.131209677295109, −7.606789525861226, −7.041246399696004, −6.569597994648543, −6.007009670683534, −5.341072389062664, −4.920759846676297, −4.201277337644628, −3.878281200732706, −3.025081492597777, −2.545408789743150, −1.656147103184717, −1.190646361006802, 0,
1.190646361006802, 1.656147103184717, 2.545408789743150, 3.025081492597777, 3.878281200732706, 4.201277337644628, 4.920759846676297, 5.341072389062664, 6.007009670683534, 6.569597994648543, 7.041246399696004, 7.606789525861226, 8.131209677295109, 8.746481668302728, 9.348776181364855, 9.876216775220684, 10.37009794157873, 10.89061842742700, 11.64790120775689, 11.88250543910161, 12.28188888481851, 12.82297868138700, 13.36957535022359, 13.99417501021348, 14.44435914318853