| L(s) = 1 | + 0.499·2-s + 0.171·3-s − 1.75·4-s + 0.0854·6-s − 3.87·7-s − 1.87·8-s − 2.97·9-s + 2.93·11-s − 0.299·12-s + 0.983·13-s − 1.93·14-s + 2.56·16-s − 1.48·18-s + 2.56·19-s − 0.663·21-s + 1.46·22-s + 3.00·23-s − 0.320·24-s + 0.491·26-s − 1.02·27-s + 6.78·28-s − 0.549·29-s − 7.55·31-s + 5.02·32-s + 0.502·33-s + 5.20·36-s + 9.84·37-s + ⋯ |
| L(s) = 1 | + 0.352·2-s + 0.0988·3-s − 0.875·4-s + 0.0349·6-s − 1.46·7-s − 0.661·8-s − 0.990·9-s + 0.884·11-s − 0.0865·12-s + 0.272·13-s − 0.516·14-s + 0.641·16-s − 0.349·18-s + 0.589·19-s − 0.144·21-s + 0.312·22-s + 0.626·23-s − 0.0654·24-s + 0.0962·26-s − 0.196·27-s + 1.28·28-s − 0.102·29-s − 1.35·31-s + 0.888·32-s + 0.0874·33-s + 0.866·36-s + 1.61·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.499T + 2T^{2} \) |
| 3 | \( 1 - 0.171T + 3T^{2} \) |
| 7 | \( 1 + 3.87T + 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 13 | \( 1 - 0.983T + 13T^{2} \) |
| 19 | \( 1 - 2.56T + 19T^{2} \) |
| 23 | \( 1 - 3.00T + 23T^{2} \) |
| 29 | \( 1 + 0.549T + 29T^{2} \) |
| 31 | \( 1 + 7.55T + 31T^{2} \) |
| 37 | \( 1 - 9.84T + 37T^{2} \) |
| 41 | \( 1 - 7.40T + 41T^{2} \) |
| 43 | \( 1 - 8.27T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 3.58T + 53T^{2} \) |
| 59 | \( 1 - 0.305T + 59T^{2} \) |
| 61 | \( 1 + 6.80T + 61T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 + 3.60T + 71T^{2} \) |
| 73 | \( 1 - 5.01T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 8.12T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 4.52T + 97T^{2} \) |
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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
−7.54391818468794712957359263584, −6.74508612474956786148519496787, −5.87099915934513582829024734743, −5.76024234589847308477892581575, −4.62136574920984009532582555643, −3.84124847785344265337283861868, −3.27928143359551965513272628421, −2.63935925775836193007545257251, −1.04682904165142744712515984622, 0,
1.04682904165142744712515984622, 2.63935925775836193007545257251, 3.27928143359551965513272628421, 3.84124847785344265337283861868, 4.62136574920984009532582555643, 5.76024234589847308477892581575, 5.87099915934513582829024734743, 6.74508612474956786148519496787, 7.54391818468794712957359263584
