| L(s) = 1 | − 2-s − 4-s + 3·8-s + 11-s − 2·13-s − 16-s − 2·17-s − 4·19-s − 22-s + 2·26-s − 6·29-s − 5·32-s + 2·34-s − 6·37-s + 4·38-s − 6·41-s + 4·43-s − 44-s + 2·52-s − 2·53-s + 6·58-s + 4·59-s − 6·61-s + 7·64-s − 12·67-s + 2·68-s + 10·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 0.301·11-s − 0.554·13-s − 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.213·22-s + 0.392·26-s − 1.11·29-s − 0.883·32-s + 0.342·34-s − 0.986·37-s + 0.648·38-s − 0.937·41-s + 0.609·43-s − 0.150·44-s + 0.277·52-s − 0.274·53-s + 0.787·58-s + 0.520·59-s − 0.768·61-s + 7/8·64-s − 1.46·67-s + 0.242·68-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3567864931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3567864931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.46250437202264, −13.11450799893547, −12.67090342143540, −12.04064479615765, −11.65836509411152, −10.87040503006370, −10.60681700239127, −10.18289193343472, −9.437555074593383, −9.223691493220172, −8.750807088547290, −8.195452327374550, −7.735834999396096, −7.198324896204741, −6.702130997122133, −6.110323841114574, −5.415114460848553, −4.881567904492192, −4.426767963135368, −3.799510671788503, −3.313334505279603, −2.291786638030129, −1.893976338934957, −1.138425336407668, −0.2228641817187375,
0.2228641817187375, 1.138425336407668, 1.893976338934957, 2.291786638030129, 3.313334505279603, 3.799510671788503, 4.426767963135368, 4.881567904492192, 5.415114460848553, 6.110323841114574, 6.702130997122133, 7.198324896204741, 7.735834999396096, 8.195452327374550, 8.750807088547290, 9.223691493220172, 9.437555074593383, 10.18289193343472, 10.60681700239127, 10.87040503006370, 11.65836509411152, 12.04064479615765, 12.67090342143540, 13.11450799893547, 13.46250437202264
