| L(s) = 1 | + 4-s + 5-s + 7-s − 11-s − 13-s + 16-s − 17-s + 20-s + 25-s + 28-s + 2·29-s + 35-s − 44-s − 47-s + 49-s − 52-s − 55-s + 64-s − 65-s − 68-s − 71-s − 73-s − 77-s − 79-s + 80-s − 83-s − 85-s + ⋯ |
| L(s) = 1 | + 4-s + 5-s + 7-s − 11-s − 13-s + 16-s − 17-s + 20-s + 25-s + 28-s + 2·29-s + 35-s − 44-s − 47-s + 49-s − 52-s − 55-s + 64-s − 65-s − 68-s − 71-s − 73-s − 77-s − 79-s + 80-s − 83-s − 85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2835 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2835 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.909390026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909390026\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−8.852694057485649070789950439648, −8.185270523283213932631908711798, −7.38904924362008247238335539021, −6.73073933565119863352483290299, −5.93301224456478426847068666706, −5.11492469806523406837344293354, −4.54398054949933163862247317309, −2.85418741015635243144614987831, −2.39079247718032847879221740436, −1.47213205889923248965535406673,
1.47213205889923248965535406673, 2.39079247718032847879221740436, 2.85418741015635243144614987831, 4.54398054949933163862247317309, 5.11492469806523406837344293354, 5.93301224456478426847068666706, 6.73073933565119863352483290299, 7.38904924362008247238335539021, 8.185270523283213932631908711798, 8.852694057485649070789950439648
