L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 3·8-s + 9-s − 10-s − 3·11-s + 12-s − 2·13-s − 15-s − 16-s − 3·17-s − 18-s + 2·19-s − 20-s + 3·22-s − 3·24-s + 25-s + 2·26-s − 27-s + 10·29-s + 30-s + 4·31-s − 5·32-s + 3·33-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s + 0.288·12-s − 0.554·13-s − 0.258·15-s − 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.639·22-s − 0.612·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 1.85·29-s + 0.182·30-s + 0.718·31-s − 0.883·32-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7722465962\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7722465962\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−12.26884884127260, −12.09121937725842, −11.55746433208069, −10.89294632028213, −10.35884143201571, −10.29351756453015, −9.960081628011113, −9.179568888481809, −8.920587250973600, −8.483089457299336, −7.902683959872734, −7.479868516493432, −7.036799784218792, −6.514827212749472, −5.963966536340856, −5.404758249806984, −5.011688132436267, −4.466128788754163, −4.310692382279435, −3.246753263852843, −2.850928910579322, −2.139248744934537, −1.580760037702299, −0.8936656864770710, −0.3318920043869011,
0.3318920043869011, 0.8936656864770710, 1.580760037702299, 2.139248744934537, 2.850928910579322, 3.246753263852843, 4.310692382279435, 4.466128788754163, 5.011688132436267, 5.404758249806984, 5.963966536340856, 6.514827212749472, 7.036799784218792, 7.479868516493432, 7.902683959872734, 8.483089457299336, 8.920587250973600, 9.179568888481809, 9.960081628011113, 10.29351756453015, 10.35884143201571, 10.89294632028213, 11.55746433208069, 12.09121937725842, 12.26884884127260