L(s) = 1 | + 2-s + 0.302·3-s + 4-s + 0.302·6-s − 3.30·7-s + 8-s − 2.90·9-s − 1.69·11-s + 0.302·12-s − 3.30·13-s − 3.30·14-s + 16-s − 6.90·17-s − 2.90·18-s + 5.90·19-s − 1.00·21-s − 1.69·22-s + 23-s + 0.302·24-s − 3.30·26-s − 1.78·27-s − 3.30·28-s − 2.60·29-s − 7.90·31-s + 32-s − 0.513·33-s − 6.90·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.174·3-s + 0.5·4-s + 0.123·6-s − 1.24·7-s + 0.353·8-s − 0.969·9-s − 0.511·11-s + 0.0874·12-s − 0.916·13-s − 0.882·14-s + 0.250·16-s − 1.67·17-s − 0.685·18-s + 1.35·19-s − 0.218·21-s − 0.361·22-s + 0.208·23-s + 0.0618·24-s − 0.647·26-s − 0.344·27-s − 0.624·28-s − 0.483·29-s − 1.42·31-s + 0.176·32-s − 0.0894·33-s − 1.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 0.302T + 3T^{2} \) |
| 7 | \( 1 + 3.30T + 7T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 + 6.90T + 17T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 29 | \( 1 + 2.60T + 29T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 0.908T + 41T^{2} \) |
| 43 | \( 1 - 9.21T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 3.39T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 - 5.81T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 6.30T + 97T^{2} \) |
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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
−9.340803057304897498554607170347, −8.734534460783700156842579658814, −7.40895269131913922781407732370, −6.93458094345463377609827273747, −5.81034043323570385534607923940, −5.24746536813923552644599284364, −4.00002778728647914320899627752, −3.04004680587820482871752922772, −2.32756097422350121058519862216, 0,
2.32756097422350121058519862216, 3.04004680587820482871752922772, 4.00002778728647914320899627752, 5.24746536813923552644599284364, 5.81034043323570385534607923940, 6.93458094345463377609827273747, 7.40895269131913922781407732370, 8.734534460783700156842579658814, 9.340803057304897498554607170347