| 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 + 14-s + 2·15-s + 16-s − 6·17-s − 18-s + 4·19-s + 20-s − 2·21-s − 6·23-s − 2·24-s + 25-s − 4·27-s − 28-s + 2·29-s − 2·30-s − 32-s + 6·34-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 + 0.267·14-s + 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.436·21-s − 1.25·23-s − 0.408·24-s + 1/5·25-s − 0.769·27-s − 0.188·28-s + 0.371·29-s − 0.365·30-s − 0.176·32-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.43988652268202, −13.95466638648963, −13.53467263411777, −13.00887260931343, −12.56429837711289, −11.84775958436961, −11.16805015635774, −11.03372907337193, −9.961020170302105, −9.817744695169343, −9.393147939130652, −8.685226755844498, −8.546521083677824, −7.721600153376872, −7.488393719977544, −6.722804510383854, −6.119054381789373, −5.785809671137604, −4.813887627846912, −4.159719780615718, −3.562813259246847, −2.757804641857639, −2.466265116623955, −1.848278861814404, −0.9819846335113499, 0,
0.9819846335113499, 1.848278861814404, 2.466265116623955, 2.757804641857639, 3.562813259246847, 4.159719780615718, 4.813887627846912, 5.785809671137604, 6.119054381789373, 6.722804510383854, 7.488393719977544, 7.721600153376872, 8.546521083677824, 8.685226755844498, 9.393147939130652, 9.817744695169343, 9.961020170302105, 11.03372907337193, 11.16805015635774, 11.84775958436961, 12.56429837711289, 13.00887260931343, 13.53467263411777, 13.95466638648963, 14.43988652268202
