| L(s) = 1 | − 4-s − 7-s + 16-s + 25-s + 28-s − 2·37-s − 2·43-s + 49-s − 64-s + 2·67-s + 2·79-s − 100-s − 2·109-s − 112-s + ⋯ |
| L(s) = 1 | − 4-s − 7-s + 16-s + 25-s + 28-s − 2·37-s − 2·43-s + 49-s − 64-s + 2·67-s + 2·79-s − 100-s − 2·109-s − 112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4040860762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4040860762\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−15.18997093836620602460637582619, −13.99915090221690521549417434165, −13.10575814689347063007868597189, −12.19346500594903014566864998433, −10.47732338749126199986584579156, −9.471054764067809621406584360430, −8.420547362205116289934079288794, −6.74489112464479961545435616225, −5.16255973701201100775027286861, −3.51328813313495541508690022552,
3.51328813313495541508690022552, 5.16255973701201100775027286861, 6.74489112464479961545435616225, 8.420547362205116289934079288794, 9.471054764067809621406584360430, 10.47732338749126199986584579156, 12.19346500594903014566864998433, 13.10575814689347063007868597189, 13.99915090221690521549417434165, 15.18997093836620602460637582619
