| L(s) = 1 | + 3-s − 7-s + 11-s + 13-s − 21-s + 23-s + 25-s − 27-s + 31-s + 33-s − 37-s + 39-s − 41-s + 47-s + 49-s − 61-s + 67-s + 69-s − 73-s + 75-s − 77-s + 79-s − 81-s + 2·89-s − 91-s + 93-s − 97-s + ⋯ |
| L(s) = 1 | + 3-s − 7-s + 11-s + 13-s − 21-s + 23-s + 25-s − 27-s + 31-s + 33-s − 37-s + 39-s − 41-s + 47-s + 49-s − 61-s + 67-s + 69-s − 73-s + 75-s − 77-s + 79-s − 81-s + 2·89-s − 91-s + 93-s − 97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.685999142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.685999142\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 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 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−8.849373549970716944323424258382, −8.525776966805402485062876912277, −7.44116450417031792384042408310, −6.65773794856622066561832897070, −6.13258191249192359507627950107, −5.02718543717052400859687443426, −3.85325149807876038283678915907, −3.35255790598384755781314044906, −2.56285936883412218955073813438, −1.23288935504299016976400626513,
1.23288935504299016976400626513, 2.56285936883412218955073813438, 3.35255790598384755781314044906, 3.85325149807876038283678915907, 5.02718543717052400859687443426, 6.13258191249192359507627950107, 6.65773794856622066561832897070, 7.44116450417031792384042408310, 8.525776966805402485062876912277, 8.849373549970716944323424258382
