| L(s) = 1 | + 2-s + 2·3-s + 4-s − 5-s + 2·6-s − 7-s + 8-s + 9-s − 10-s + 2·12-s − 2·13-s − 14-s − 2·15-s + 16-s − 2·17-s + 18-s − 6·19-s − 20-s − 2·21-s + 6·23-s + 2·24-s + 25-s − 2·26-s − 4·27-s − 28-s − 4·29-s − 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s − 0.554·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.436·21-s + 1.25·23-s + 0.408·24-s + 1/5·25-s − 0.392·26-s − 0.769·27-s − 0.188·28-s − 0.742·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.56249007342954795202770288168, −6.71661484563657249081930337387, −6.20085569107754523744484230066, −5.17592566352570205119085052185, −4.45571685681131296945082979290, −3.83648917823894344436117584024, −3.02654794616741752365659974603, −2.56624929779305273228396802158, −1.64079608546800263227732505868, 0,
1.64079608546800263227732505868, 2.56624929779305273228396802158, 3.02654794616741752365659974603, 3.83648917823894344436117584024, 4.45571685681131296945082979290, 5.17592566352570205119085052185, 6.20085569107754523744484230066, 6.71661484563657249081930337387, 7.56249007342954795202770288168
