L(s) = 1 | − 2-s + 4-s + 5-s − 3·7-s − 8-s − 10-s + 11-s − 6·13-s + 3·14-s + 16-s − 2·17-s + 8·19-s + 20-s − 22-s + 4·23-s + 25-s + 6·26-s − 3·28-s − 5·29-s − 32-s + 2·34-s − 3·35-s + 4·37-s − 8·38-s − 40-s − 6·41-s + 2·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.13·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s − 1.66·13-s + 0.801·14-s + 1/4·16-s − 0.485·17-s + 1.83·19-s + 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s + 1.17·26-s − 0.566·28-s − 0.928·29-s − 0.176·32-s + 0.342·34-s − 0.507·35-s + 0.657·37-s − 1.29·38-s − 0.158·40-s − 0.937·41-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8183566128\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8183566128\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.95426116051174, −13.25653147563419, −12.92862050925402, −12.46272812708113, −11.85419885553932, −11.39527514846668, −10.97737536187156, −10.03422571885111, −9.889848979042908, −9.442590267487159, −9.215464608705215, −8.414669448662089, −7.771193671962873, −7.271363497981884, −6.815985377298697, −6.458317439623363, −5.670326957818115, −5.171624018042467, −4.681310127554133, −3.667327903547132, −3.142237434695209, −2.674845363876730, −1.967344932450916, −1.207038059872351, −0.3392179376995161,
0.3392179376995161, 1.207038059872351, 1.967344932450916, 2.674845363876730, 3.142237434695209, 3.667327903547132, 4.681310127554133, 5.171624018042467, 5.670326957818115, 6.458317439623363, 6.815985377298697, 7.271363497981884, 7.771193671962873, 8.414669448662089, 9.215464608705215, 9.442590267487159, 9.889848979042908, 10.03422571885111, 10.97737536187156, 11.39527514846668, 11.85419885553932, 12.46272812708113, 12.92862050925402, 13.25653147563419, 13.95426116051174