L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 2·7-s − 8-s + 9-s − 10-s + 12-s + 4·13-s − 2·14-s + 15-s + 16-s − 17-s − 18-s + 6·19-s + 20-s + 2·21-s − 2·23-s − 24-s + 25-s − 4·26-s + 27-s + 2·28-s + 6·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.10·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.436·21-s − 0.417·23-s − 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.377·28-s + 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.317644273\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.317644273\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.23288751895821, −14.00289843019659, −13.18323140511799, −12.98961067564822, −12.14359060588845, −11.52497008464113, −11.30795580322942, −10.62850342145107, −10.04644524700497, −9.737423134844056, −8.976737662749613, −8.712660277930226, −8.103364749536346, −7.714960728173168, −7.120645139792748, −6.407632798213393, −6.048324987931907, −5.142627904391266, −4.832789073923590, −3.824688681440930, −3.354931804225680, −2.664391998686408, −1.859856659163441, −1.439095510848318, −0.6974897829806309,
0.6974897829806309, 1.439095510848318, 1.859856659163441, 2.664391998686408, 3.354931804225680, 3.824688681440930, 4.832789073923590, 5.142627904391266, 6.048324987931907, 6.407632798213393, 7.120645139792748, 7.714960728173168, 8.103364749536346, 8.712660277930226, 8.976737662749613, 9.737423134844056, 10.04644524700497, 10.62850342145107, 11.30795580322942, 11.52497008464113, 12.14359060588845, 12.98961067564822, 13.18323140511799, 14.00289843019659, 14.23288751895821