| L(s) = 1 | + 2·3-s − 5-s + 9-s + 2·11-s + 2·13-s − 2·15-s + 6·17-s + 6·19-s + 25-s − 4·27-s + 10·29-s − 8·31-s + 4·33-s + 2·37-s + 4·39-s − 6·41-s − 2·43-s − 45-s − 12·47-s − 7·49-s + 12·51-s + 10·53-s − 2·55-s + 12·57-s − 6·59-s − 6·61-s − 2·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.516·15-s + 1.45·17-s + 1.37·19-s + 1/5·25-s − 0.769·27-s + 1.85·29-s − 1.43·31-s + 0.696·33-s + 0.328·37-s + 0.640·39-s − 0.937·41-s − 0.304·43-s − 0.149·45-s − 1.75·47-s − 49-s + 1.68·51-s + 1.37·53-s − 0.269·55-s + 1.58·57-s − 0.781·59-s − 0.768·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.146865549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.146865549\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−10.39102304097340406960115303065, −9.546865557077499447722022189862, −8.785640940627923079947670161672, −8.003973673628002575380076993759, −7.34544995247272548117733891306, −6.13680123590138963079680336084, −4.93901359493313369458876428491, −3.56314650163491707611542821048, −3.10625138730091820603192538278, −1.41580456294010257565810162991,
1.41580456294010257565810162991, 3.10625138730091820603192538278, 3.56314650163491707611542821048, 4.93901359493313369458876428491, 6.13680123590138963079680336084, 7.34544995247272548117733891306, 8.003973673628002575380076993759, 8.785640940627923079947670161672, 9.546865557077499447722022189862, 10.39102304097340406960115303065
