| L(s) = 1 | + 4-s − 5-s + 7-s + 9-s − 11-s − 2·13-s + 16-s + 17-s + 19-s − 20-s − 23-s + 28-s + 29-s − 31-s − 35-s + 36-s − 37-s − 41-s − 44-s − 45-s − 2·52-s + 55-s − 59-s + 63-s + 64-s + 2·65-s + 68-s + ⋯ |
| L(s) = 1 | + 4-s − 5-s + 7-s + 9-s − 11-s − 2·13-s + 16-s + 17-s + 19-s − 20-s − 23-s + 28-s + 29-s − 31-s − 35-s + 36-s − 37-s − 41-s − 44-s − 45-s − 2·52-s + 55-s − 59-s + 63-s + 64-s + 2·65-s + 68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9658560119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9658560119\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 + T )^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 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
−11.48102960621235442156169355230, −10.35245861439623828614753841311, −9.901442526884833753787951019147, −8.112927732564344217882160023371, −7.55340885121323948880340960318, −7.15454564452373175045583083920, −5.45077398517606168828083141192, −4.62692691740037057523082253517, −3.21226091457868276887990528985, −1.86988117279222023205677327092,
1.86988117279222023205677327092, 3.21226091457868276887990528985, 4.62692691740037057523082253517, 5.45077398517606168828083141192, 7.15454564452373175045583083920, 7.55340885121323948880340960318, 8.112927732564344217882160023371, 9.901442526884833753787951019147, 10.35245861439623828614753841311, 11.48102960621235442156169355230
