L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 11-s − 12-s − 4·13-s + 16-s + 2·17-s + 18-s − 19-s − 22-s − 2·23-s − 24-s − 4·26-s − 27-s + 5·29-s − 5·31-s + 32-s + 33-s + 2·34-s + 36-s + 6·37-s − 38-s + 4·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.229·19-s − 0.213·22-s − 0.417·23-s − 0.204·24-s − 0.784·26-s − 0.192·27-s + 0.928·29-s − 0.898·31-s + 0.176·32-s + 0.174·33-s + 0.342·34-s + 1/6·36-s + 0.986·37-s − 0.162·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.483358889\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.483358889\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.27724280861945, −12.86701686169014, −12.50387565133114, −12.01225523234601, −11.63020521893778, −11.08819276384219, −10.65993157812278, −10.06021446973439, −9.710587822363597, −9.254243337841629, −8.297394637495901, −7.971491965230399, −7.490127390657308, −6.755676453687450, −6.556563001689977, −5.865999140405612, −5.300053316010899, −4.981147960501380, −4.396735486199498, −3.904417165662007, −3.072790740794308, −2.727693668910354, −1.894401355002903, −1.358935469474041, −0.3233841675508071,
0.3233841675508071, 1.358935469474041, 1.894401355002903, 2.727693668910354, 3.072790740794308, 3.904417165662007, 4.396735486199498, 4.981147960501380, 5.300053316010899, 5.865999140405612, 6.556563001689977, 6.755676453687450, 7.490127390657308, 7.971491965230399, 8.297394637495901, 9.254243337841629, 9.710587822363597, 10.06021446973439, 10.65993157812278, 11.08819276384219, 11.63020521893778, 12.01225523234601, 12.50387565133114, 12.86701686169014, 13.27724280861945