L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 4·7-s + 8-s + 9-s + 10-s + 4·11-s + 12-s + 13-s − 4·14-s + 15-s + 16-s + 6·17-s + 18-s − 4·19-s + 20-s − 4·21-s + 4·22-s + 8·23-s + 24-s + 25-s + 26-s + 27-s − 4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.872·21-s + 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.768646726\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.768646726\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.54419824487900, −12.34489263677785, −11.71202445651609, −11.07080594290615, −10.77957666284352, −10.11510565568751, −9.711193962885440, −9.400037366221376, −8.972423904127835, −8.477819900675225, −7.804065339968828, −7.359893593070648, −6.778296255953862, −6.352134669459031, −6.169688210665004, −5.627503333417147, −4.870393717559088, −4.407096717659160, −3.894342709831200, −3.338989522255651, −2.892379513817592, −2.724717929364942, −1.721297815703515, −1.210989996957617, −0.6378123912784338,
0.6378123912784338, 1.210989996957617, 1.721297815703515, 2.724717929364942, 2.892379513817592, 3.338989522255651, 3.894342709831200, 4.407096717659160, 4.870393717559088, 5.627503333417147, 6.169688210665004, 6.352134669459031, 6.778296255953862, 7.359893593070648, 7.804065339968828, 8.477819900675225, 8.972423904127835, 9.400037366221376, 9.711193962885440, 10.11510565568751, 10.77957666284352, 11.07080594290615, 11.71202445651609, 12.34489263677785, 12.54419824487900