L(s) = 1 | − 3-s − 3·5-s − 7-s − 2·9-s − 11-s − 13-s + 3·15-s + 21-s − 7·23-s + 4·25-s + 5·27-s − 7·31-s + 33-s + 3·35-s − 7·37-s + 39-s − 4·41-s − 2·43-s + 6·45-s + 49-s − 10·53-s + 3·55-s − 9·59-s − 12·61-s + 2·63-s + 3·65-s − 7·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s − 2/3·9-s − 0.301·11-s − 0.277·13-s + 0.774·15-s + 0.218·21-s − 1.45·23-s + 4/5·25-s + 0.962·27-s − 1.25·31-s + 0.174·33-s + 0.507·35-s − 1.15·37-s + 0.160·39-s − 0.624·41-s − 0.304·43-s + 0.894·45-s + 1/7·49-s − 1.37·53-s + 0.404·55-s − 1.17·59-s − 1.53·61-s + 0.251·63-s + 0.372·65-s − 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−7.20000398617395120502475584076, −6.44201991818607500453659310579, −5.75449841207006370633424147447, −5.04606909140504745327308384069, −4.27944952390062330748873882071, −3.52855862315579338609758792819, −2.89640095692064093470088457577, −1.68280721367039493739207933996, 0, 0,
1.68280721367039493739207933996, 2.89640095692064093470088457577, 3.52855862315579338609758792819, 4.27944952390062330748873882071, 5.04606909140504745327308384069, 5.75449841207006370633424147447, 6.44201991818607500453659310579, 7.20000398617395120502475584076