| L(s) = 1 | + 1.25·2-s + 0.772·3-s − 0.422·4-s + 5-s + 0.969·6-s − 3.10·7-s − 3.04·8-s − 2.40·9-s + 1.25·10-s + 11-s − 0.326·12-s − 3.91·13-s − 3.89·14-s + 0.772·15-s − 2.97·16-s − 2.90·17-s − 3.01·18-s − 19-s − 0.422·20-s − 2.39·21-s + 1.25·22-s − 3.98·23-s − 2.34·24-s + 25-s − 4.92·26-s − 4.17·27-s + 1.31·28-s + ⋯ |
| L(s) = 1 | + 0.888·2-s + 0.445·3-s − 0.211·4-s + 0.447·5-s + 0.395·6-s − 1.17·7-s − 1.07·8-s − 0.801·9-s + 0.397·10-s + 0.301·11-s − 0.0941·12-s − 1.08·13-s − 1.04·14-s + 0.199·15-s − 0.744·16-s − 0.703·17-s − 0.711·18-s − 0.229·19-s − 0.0944·20-s − 0.522·21-s + 0.267·22-s − 0.831·23-s − 0.479·24-s + 0.200·25-s − 0.965·26-s − 0.803·27-s + 0.247·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.25T + 2T^{2} \) |
| 3 | \( 1 - 0.772T + 3T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 13 | \( 1 + 3.91T + 13T^{2} \) |
| 17 | \( 1 + 2.90T + 17T^{2} \) |
| 23 | \( 1 + 3.98T + 23T^{2} \) |
| 29 | \( 1 - 3.59T + 29T^{2} \) |
| 31 | \( 1 - 2.66T + 31T^{2} \) |
| 37 | \( 1 - 7.20T + 37T^{2} \) |
| 41 | \( 1 - 9.16T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 3.48T + 47T^{2} \) |
| 53 | \( 1 + 4.59T + 53T^{2} \) |
| 59 | \( 1 + 2.68T + 59T^{2} \) |
| 61 | \( 1 + 5.05T + 61T^{2} \) |
| 67 | \( 1 - 3.57T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + 8.68T + 73T^{2} \) |
| 79 | \( 1 + 6.39T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 0.379T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−9.521666859834573405441177967615, −8.848932097720388741914041713496, −7.911490130900882689178479962642, −6.53781419535737698803520144889, −6.13989333168634237771281483518, −5.08234996227703061469335299396, −4.16167959564150812234345353478, −3.11170383921793773547161539835, −2.44681490263965264973480278434, 0,
2.44681490263965264973480278434, 3.11170383921793773547161539835, 4.16167959564150812234345353478, 5.08234996227703061469335299396, 6.13989333168634237771281483518, 6.53781419535737698803520144889, 7.911490130900882689178479962642, 8.848932097720388741914041713496, 9.521666859834573405441177967615
