L(s) = 1 | + 3-s + 3·7-s + 9-s + 3·11-s − 13-s + 17-s + 6·19-s + 3·21-s + 5·23-s + 27-s + 6·29-s + 2·31-s + 3·33-s + 7·37-s − 39-s + 3·41-s − 8·43-s + 2·47-s + 2·49-s + 51-s − 53-s + 6·57-s + 15·61-s + 3·63-s + 12·67-s + 5·69-s + 5·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 0.242·17-s + 1.37·19-s + 0.654·21-s + 1.04·23-s + 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.522·33-s + 1.15·37-s − 0.160·39-s + 0.468·41-s − 1.21·43-s + 0.291·47-s + 2/7·49-s + 0.140·51-s − 0.137·53-s + 0.794·57-s + 1.92·61-s + 0.377·63-s + 1.46·67-s + 0.601·69-s + 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.487906728\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.487906728\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−14.35740416651878, −13.89479210318369, −13.37941862907299, −12.80059284736153, −12.15121711327821, −11.72657194906931, −11.30196872255963, −10.83211792962831, −9.997980969259586, −9.667666197912910, −9.155424240026808, −8.511188575525852, −8.078767494642429, −7.682706120775479, −6.853563300236841, −6.706729711613941, −5.689351477999326, −5.136225803676017, −4.684575640911545, −4.039696759055062, −3.391784587756156, −2.753935876090213, −2.104006087223902, −1.193734395518045, −0.9249848433628054,
0.9249848433628054, 1.193734395518045, 2.104006087223902, 2.753935876090213, 3.391784587756156, 4.039696759055062, 4.684575640911545, 5.136225803676017, 5.689351477999326, 6.706729711613941, 6.853563300236841, 7.682706120775479, 8.078767494642429, 8.511188575525852, 9.155424240026808, 9.667666197912910, 9.997980969259586, 10.83211792962831, 11.30196872255963, 11.72657194906931, 12.15121711327821, 12.80059284736153, 13.37941862907299, 13.89479210318369, 14.35740416651878