| L(s) = 1 | + 3-s − 5-s − 2·9-s + 6·11-s + 2·13-s − 15-s − 6·17-s + 8·19-s + 3·23-s + 25-s − 5·27-s + 3·29-s + 2·31-s + 6·33-s + 8·37-s + 2·39-s − 3·41-s + 5·43-s + 2·45-s − 6·51-s + 12·53-s − 6·55-s + 8·57-s − 61-s − 2·65-s − 7·67-s + 3·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s + 1.80·11-s + 0.554·13-s − 0.258·15-s − 1.45·17-s + 1.83·19-s + 0.625·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s + 0.359·31-s + 1.04·33-s + 1.31·37-s + 0.320·39-s − 0.468·41-s + 0.762·43-s + 0.298·45-s − 0.840·51-s + 1.64·53-s − 0.809·55-s + 1.05·57-s − 0.128·61-s − 0.248·65-s − 0.855·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.916226135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.916226135\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.737338981474385039931332756559, −8.985500970998463353247074507528, −8.599551883441898492053947482088, −7.49478627956930010686558944624, −6.68749734040846354911986782288, −5.79771817316828836924001759466, −4.49002124505733078607521968576, −3.65906960024022852291238422490, −2.70727114515163114275867183178, −1.14605640894775581861792784282,
1.14605640894775581861792784282, 2.70727114515163114275867183178, 3.65906960024022852291238422490, 4.49002124505733078607521968576, 5.79771817316828836924001759466, 6.68749734040846354911986782288, 7.49478627956930010686558944624, 8.599551883441898492053947482088, 8.985500970998463353247074507528, 9.737338981474385039931332756559
