| L(s) = 1 | − 2.30·2-s + 1.87·3-s + 3.31·4-s − 4.32·6-s − 1.54·7-s − 3.03·8-s + 0.525·9-s + 4.61·11-s + 6.22·12-s + 3.51·13-s + 3.56·14-s + 0.366·16-s − 1.21·18-s − 6.62·19-s − 2.90·21-s − 10.6·22-s + 2.59·23-s − 5.70·24-s − 8.11·26-s − 4.64·27-s − 5.12·28-s − 0.979·29-s + 1.22·31-s + 5.22·32-s + 8.67·33-s + 1.74·36-s − 0.407·37-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.08·3-s + 1.65·4-s − 1.76·6-s − 0.584·7-s − 1.07·8-s + 0.175·9-s + 1.39·11-s + 1.79·12-s + 0.975·13-s + 0.952·14-s + 0.0916·16-s − 0.285·18-s − 1.51·19-s − 0.633·21-s − 2.27·22-s + 0.541·23-s − 1.16·24-s − 1.59·26-s − 0.894·27-s − 0.968·28-s − 0.181·29-s + 0.219·31-s + 0.923·32-s + 1.50·33-s + 0.290·36-s − 0.0670·37-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 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 - 1.87T + 3T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 11 | \( 1 - 4.61T + 11T^{2} \) |
| 13 | \( 1 - 3.51T + 13T^{2} \) |
| 19 | \( 1 + 6.62T + 19T^{2} \) |
| 23 | \( 1 - 2.59T + 23T^{2} \) |
| 29 | \( 1 + 0.979T + 29T^{2} \) |
| 31 | \( 1 - 1.22T + 31T^{2} \) |
| 37 | \( 1 + 0.407T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 - 3.99T + 47T^{2} \) |
| 53 | \( 1 + 1.75T + 53T^{2} \) |
| 59 | \( 1 - 1.56T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 8.84T + 67T^{2} \) |
| 71 | \( 1 + 1.29T + 71T^{2} \) |
| 73 | \( 1 + 9.77T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 8.62T + 89T^{2} \) |
| 97 | \( 1 + 5.92T + 97T^{2} \) |
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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
−7.966067295797871977614811548414, −6.94319966583268406188638258675, −6.61546685062811788394416606953, −5.89470577278806528462047760408, −4.44788560512365127136286903666, −3.64678424917222717876681475359, −2.96200126624706109823356445082, −1.95963502762029590923170102413, −1.32778051353071546491843175399, 0,
1.32778051353071546491843175399, 1.95963502762029590923170102413, 2.96200126624706109823356445082, 3.64678424917222717876681475359, 4.44788560512365127136286903666, 5.89470577278806528462047760408, 6.61546685062811788394416606953, 6.94319966583268406188638258675, 7.966067295797871977614811548414
