| L(s) = 1 | − 3-s + 9-s − 4·11-s + 2·13-s − 2·17-s − 4·19-s + 4·23-s − 5·25-s − 27-s − 2·31-s + 4·33-s − 2·37-s − 2·39-s − 2·41-s − 8·43-s + 4·47-s − 7·49-s + 2·51-s − 4·53-s + 4·57-s + 6·59-s + 2·61-s + 8·67-s − 4·69-s − 8·71-s − 4·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s − 25-s − 0.192·27-s − 0.359·31-s + 0.696·33-s − 0.328·37-s − 0.320·39-s − 0.312·41-s − 1.21·43-s + 0.583·47-s − 49-s + 0.280·51-s − 0.549·53-s + 0.529·57-s + 0.781·59-s + 0.256·61-s + 0.977·67-s − 0.481·69-s − 0.949·71-s − 0.468·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8098656709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8098656709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.68903233868068, −15.19717795332085, −14.73648653131162, −13.84181278827466, −13.38408231509319, −12.90402867497079, −12.50940139759839, −11.65701443304247, −11.22190815344571, −10.73663281184712, −10.19328420716356, −9.664559604680948, −8.827574989864024, −8.342956415466446, −7.743022101648215, −7.014461754343138, −6.491237504131383, −5.812852273223089, −5.230421781629605, −4.664003264451779, −3.910660074077356, −3.159395988243021, −2.286430068062999, −1.569427027872612, −0.3763697915626634,
0.3763697915626634, 1.569427027872612, 2.286430068062999, 3.159395988243021, 3.910660074077356, 4.664003264451779, 5.230421781629605, 5.812852273223089, 6.491237504131383, 7.014461754343138, 7.743022101648215, 8.342956415466446, 8.827574989864024, 9.664559604680948, 10.19328420716356, 10.73663281184712, 11.22190815344571, 11.65701443304247, 12.50940139759839, 12.90402867497079, 13.38408231509319, 13.84181278827466, 14.73648653131162, 15.19717795332085, 15.68903233868068
