| L(s) = 1 | + 2·3-s − 5-s + 9-s − 4·11-s − 2·13-s − 2·15-s + 2·19-s + 4·23-s + 25-s − 4·27-s + 10·29-s + 4·31-s − 8·33-s − 2·37-s − 4·39-s + 12·41-s + 4·43-s − 45-s + 4·47-s + 2·53-s + 4·55-s + 4·57-s + 10·59-s − 6·61-s + 2·65-s − 4·67-s + 8·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s + 0.458·19-s + 0.834·23-s + 1/5·25-s − 0.769·27-s + 1.85·29-s + 0.718·31-s − 1.39·33-s − 0.328·37-s − 0.640·39-s + 1.87·41-s + 0.609·43-s − 0.149·45-s + 0.583·47-s + 0.274·53-s + 0.539·55-s + 0.529·57-s + 1.30·59-s − 0.768·61-s + 0.248·65-s − 0.488·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.352750387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.352750387\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.421727505094491917610175539249, −7.76782600025750632489781147968, −7.36559599206562738108486425600, −6.36493101724087924155983701559, −5.31569537402236859783928143724, −4.64720234684628632219427671710, −3.69120846358868746641322083543, −2.74761659507906835224359441772, −2.46375709317170479810809074894, −0.821430424853002211380336319600,
0.821430424853002211380336319600, 2.46375709317170479810809074894, 2.74761659507906835224359441772, 3.69120846358868746641322083543, 4.64720234684628632219427671710, 5.31569537402236859783928143724, 6.36493101724087924155983701559, 7.36559599206562738108486425600, 7.76782600025750632489781147968, 8.421727505094491917610175539249
