L(s) = 1 | + 3-s − 4-s − 2·7-s + 9-s − 12-s + 16-s − 2·21-s − 25-s + 27-s + 2·28-s − 36-s + 37-s + 48-s + 3·49-s − 2·63-s − 64-s − 2·67-s − 2·73-s − 75-s + 81-s + 2·84-s + 100-s − 108-s + 111-s − 2·112-s + ⋯ |
L(s) = 1 | + 3-s − 4-s − 2·7-s + 9-s − 12-s + 16-s − 2·21-s − 25-s + 27-s + 2·28-s − 36-s + 37-s + 48-s + 3·49-s − 2·63-s − 64-s − 2·67-s − 2·73-s − 75-s + 81-s + 2·84-s + 100-s − 108-s + 111-s − 2·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5829903212\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5829903212\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 2 | \( 1 + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 + T )^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 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
−13.61912087983946106403182530122, −13.17811219851589701555407411251, −12.30100264888277957817024823027, −10.18777066568050913245572485824, −9.568599253291209053315267014723, −8.774765372840556032518468776584, −7.44584650258568254295138388593, −6.05257527874793912859985097248, −4.14852411809580231378989051158, −3.07636964433466202517570300634,
3.07636964433466202517570300634, 4.14852411809580231378989051158, 6.05257527874793912859985097248, 7.44584650258568254295138388593, 8.774765372840556032518468776584, 9.568599253291209053315267014723, 10.18777066568050913245572485824, 12.30100264888277957817024823027, 13.17811219851589701555407411251, 13.61912087983946106403182530122