L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s + 2·11-s − 13-s − 2·14-s + 16-s + 4·17-s − 3·19-s − 2·22-s + 6·23-s + 26-s + 2·28-s + 29-s − 32-s − 4·34-s + 37-s + 3·38-s + 5·41-s − 4·43-s + 2·44-s − 6·46-s + 3·47-s − 3·49-s − 52-s + 7·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.603·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s − 0.688·19-s − 0.426·22-s + 1.25·23-s + 0.196·26-s + 0.377·28-s + 0.185·29-s − 0.176·32-s − 0.685·34-s + 0.164·37-s + 0.486·38-s + 0.780·41-s − 0.609·43-s + 0.301·44-s − 0.884·46-s + 0.437·47-s − 3/7·49-s − 0.138·52-s + 0.961·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.704715415\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.704715415\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.146345226687399473657867685730, −7.48925470053962988064826859992, −6.85801508869846162765894016225, −6.07929666098200612473526970645, −5.25701557861844685905418699483, −4.51411243850546494920785318552, −3.56240686766639342498732309914, −2.63405004288942389064879505275, −1.66545928258095207069977215397, −0.811994789339985825942040221242,
0.811994789339985825942040221242, 1.66545928258095207069977215397, 2.63405004288942389064879505275, 3.56240686766639342498732309914, 4.51411243850546494920785318552, 5.25701557861844685905418699483, 6.07929666098200612473526970645, 6.85801508869846162765894016225, 7.48925470053962988064826859992, 8.146345226687399473657867685730