L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s − 2·9-s + 12-s + 13-s − 14-s + 16-s − 6·17-s + 2·18-s − 4·19-s + 21-s − 2·23-s − 24-s − 26-s − 5·27-s + 28-s − 6·29-s − 10·31-s − 32-s + 6·34-s − 2·36-s + 2·37-s + 4·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.471·18-s − 0.917·19-s + 0.218·21-s − 0.417·23-s − 0.204·24-s − 0.196·26-s − 0.962·27-s + 0.188·28-s − 1.11·29-s − 1.79·31-s − 0.176·32-s + 1.02·34-s − 1/3·36-s + 0.328·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 293 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.36136337034264, −13.77869156309010, −13.13646750120800, −12.92159206150222, −12.21835676615487, −11.55879856704893, −11.14305515126929, −10.87082499373877, −10.37639394470409, −9.525347164896084, −9.135640414766949, −8.871284293434105, −8.351842956356244, −7.722464448214023, −7.512979454148859, −6.684212634066175, −6.193103192321271, −5.759620738165542, −4.996662456988955, −4.364145140005180, −3.741512387245979, −3.173724619157054, −2.418564587017356, −1.950043931825183, −1.459943665935949, 0, 0,
1.459943665935949, 1.950043931825183, 2.418564587017356, 3.173724619157054, 3.741512387245979, 4.364145140005180, 4.996662456988955, 5.759620738165542, 6.193103192321271, 6.684212634066175, 7.512979454148859, 7.722464448214023, 8.351842956356244, 8.871284293434105, 9.135640414766949, 9.525347164896084, 10.37639394470409, 10.87082499373877, 11.14305515126929, 11.55879856704893, 12.21835676615487, 12.92159206150222, 13.13646750120800, 13.77869156309010, 14.36136337034264