| L(s) = 1 | + 3-s + 4-s − 5-s + 7-s + 12-s − 15-s + 16-s − 20-s + 21-s + 25-s − 27-s + 28-s + 31-s − 35-s − 43-s + 48-s + 49-s − 53-s + 59-s − 60-s + 64-s − 71-s − 73-s + 75-s − 79-s − 80-s − 81-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s − 5-s + 7-s + 12-s − 15-s + 16-s − 20-s + 21-s + 25-s − 27-s + 28-s + 31-s − 35-s − 43-s + 48-s + 49-s − 53-s + 59-s − 60-s + 64-s − 71-s − 73-s + 75-s − 79-s − 80-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.954652831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.954652831\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−8.661668827720639479907630316269, −8.325965599001708071449486497222, −7.64343802754598041200362207141, −7.11286018663275475710885690122, −6.10741230778949741267887803262, −5.06190758370419477932640954714, −4.12125926868061284154249549932, −3.22910906936097877200052030190, −2.51845563174701039352660896496, −1.45593858288276049696480537024,
1.45593858288276049696480537024, 2.51845563174701039352660896496, 3.22910906936097877200052030190, 4.12125926868061284154249549932, 5.06190758370419477932640954714, 6.10741230778949741267887803262, 7.11286018663275475710885690122, 7.64343802754598041200362207141, 8.325965599001708071449486497222, 8.661668827720639479907630316269
