| L(s) = 1 | + 4-s + 9-s − 11-s + 16-s + 17-s − 23-s + 25-s − 31-s + 36-s − 37-s + 41-s + 43-s − 44-s + 49-s − 2·61-s + 64-s + 68-s − 71-s + 73-s + 79-s + 81-s − 2·83-s − 89-s − 92-s − 99-s + 100-s − 113-s + ⋯ |
| L(s) = 1 | + 4-s + 9-s − 11-s + 16-s + 17-s − 23-s + 25-s − 31-s + 36-s − 37-s + 41-s + 43-s − 44-s + 49-s − 2·61-s + 64-s + 68-s − 71-s + 73-s + 79-s + 81-s − 2·83-s − 89-s − 92-s − 99-s + 100-s − 113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.625656111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.625656111\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 233 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 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
−9.138238770019637857515807416091, −8.038455575902191871532525256564, −7.50655293335190340913384381869, −6.95972156521946188203131360537, −5.95665714033469672537204693292, −5.34484361578186438437618562053, −4.25131015883942217649791022222, −3.27003321731519927151931412343, −2.35914235718737809731244368492, −1.34360636371185014023045938662,
1.34360636371185014023045938662, 2.35914235718737809731244368492, 3.27003321731519927151931412343, 4.25131015883942217649791022222, 5.34484361578186438437618562053, 5.95665714033469672537204693292, 6.95972156521946188203131360537, 7.50655293335190340913384381869, 8.038455575902191871532525256564, 9.138238770019637857515807416091
