| L(s) = 1 | − 3-s + 3·5-s − 2·9-s + 3·11-s + 4·13-s − 3·15-s + 7·19-s + 6·23-s + 4·25-s + 5·27-s + 6·29-s + 10·31-s − 3·33-s + 2·37-s − 4·39-s + 41-s − 4·43-s − 6·45-s − 12·47-s − 6·53-s + 9·55-s − 7·57-s − 6·59-s + 13·61-s + 12·65-s − 4·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 2/3·9-s + 0.904·11-s + 1.10·13-s − 0.774·15-s + 1.60·19-s + 1.25·23-s + 4/5·25-s + 0.962·27-s + 1.11·29-s + 1.79·31-s − 0.522·33-s + 0.328·37-s − 0.640·39-s + 0.156·41-s − 0.609·43-s − 0.894·45-s − 1.75·47-s − 0.824·53-s + 1.21·55-s − 0.927·57-s − 0.781·59-s + 1.66·61-s + 1.48·65-s − 0.488·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.766700999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.766700999\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.87544414864010996144621928368, −6.75756948195720900753087314592, −6.38643181613347500889667329975, −5.85244078673054029144595273682, −5.16279932039337527499055667758, −4.54204924791500391610001749487, −3.25650099484346265388104037064, −2.81731894551521152294197303779, −1.47377143688351137743760897712, −0.973424600106608374879870448230,
0.973424600106608374879870448230, 1.47377143688351137743760897712, 2.81731894551521152294197303779, 3.25650099484346265388104037064, 4.54204924791500391610001749487, 5.16279932039337527499055667758, 5.85244078673054029144595273682, 6.38643181613347500889667329975, 6.75756948195720900753087314592, 7.87544414864010996144621928368
