| L(s) = 1 | + 2·3-s − 2·7-s + 9-s − 13-s − 5·17-s + 6·19-s − 4·21-s − 2·23-s − 4·27-s + 9·29-s − 2·31-s − 3·37-s − 2·39-s + 5·41-s − 2·47-s − 3·49-s − 10·51-s + 9·53-s + 12·57-s − 8·59-s + 6·61-s − 2·63-s + 2·67-s − 4·69-s + 12·71-s − 2·73-s + 10·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 1/3·9-s − 0.277·13-s − 1.21·17-s + 1.37·19-s − 0.872·21-s − 0.417·23-s − 0.769·27-s + 1.67·29-s − 0.359·31-s − 0.493·37-s − 0.320·39-s + 0.780·41-s − 0.291·47-s − 3/7·49-s − 1.40·51-s + 1.23·53-s + 1.58·57-s − 1.04·59-s + 0.768·61-s − 0.251·63-s + 0.244·67-s − 0.481·69-s + 1.42·71-s − 0.234·73-s + 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.579899607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.579899607\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.33726851240898, −12.64131020395410, −12.26034007342458, −11.75583140913278, −11.20692309754845, −10.71520411540833, −10.04400869338139, −9.731893728881687, −9.245274093853644, −8.910982598141161, −8.315918381099501, −7.957572482066759, −7.415027551093878, −6.744008491800977, −6.598028229676228, −5.754908276334313, −5.308533893632771, −4.608141172323179, −4.074933512249921, −3.493401896807475, −3.028396275085165, −2.542685267832726, −2.076858503554763, −1.241078697752607, −0.4206764591869321,
0.4206764591869321, 1.241078697752607, 2.076858503554763, 2.542685267832726, 3.028396275085165, 3.493401896807475, 4.074933512249921, 4.608141172323179, 5.308533893632771, 5.754908276334313, 6.598028229676228, 6.744008491800977, 7.415027551093878, 7.957572482066759, 8.315918381099501, 8.910982598141161, 9.245274093853644, 9.731893728881687, 10.04400869338139, 10.71520411540833, 11.20692309754845, 11.75583140913278, 12.26034007342458, 12.64131020395410, 13.33726851240898
