L(s) = 1 | + 2-s − 3.40·3-s + 4-s − 5-s − 3.40·6-s − 7-s + 8-s + 8.61·9-s − 10-s − 3.40·12-s − 3.69·13-s − 14-s + 3.40·15-s + 16-s − 3.63·17-s + 8.61·18-s + 2.01·19-s − 20-s + 3.40·21-s − 5.90·23-s − 3.40·24-s + 25-s − 3.69·26-s − 19.1·27-s − 28-s − 2.77·29-s + 3.40·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.96·3-s + 0.5·4-s − 0.447·5-s − 1.39·6-s − 0.377·7-s + 0.353·8-s + 2.87·9-s − 0.316·10-s − 0.984·12-s − 1.02·13-s − 0.267·14-s + 0.880·15-s + 0.250·16-s − 0.880·17-s + 2.03·18-s + 0.462·19-s − 0.223·20-s + 0.743·21-s − 1.23·23-s − 0.695·24-s + 0.200·25-s − 0.724·26-s − 3.68·27-s − 0.188·28-s − 0.515·29-s + 0.622·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7154320367\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7154320367\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 3.40T + 3T^{2} \) |
| 13 | \( 1 + 3.69T + 13T^{2} \) |
| 17 | \( 1 + 3.63T + 17T^{2} \) |
| 19 | \( 1 - 2.01T + 19T^{2} \) |
| 23 | \( 1 + 5.90T + 23T^{2} \) |
| 29 | \( 1 + 2.77T + 29T^{2} \) |
| 31 | \( 1 - 2.98T + 31T^{2} \) |
| 37 | \( 1 - 3.22T + 37T^{2} \) |
| 41 | \( 1 + 8.13T + 41T^{2} \) |
| 43 | \( 1 + 3.87T + 43T^{2} \) |
| 47 | \( 1 - 9.67T + 47T^{2} \) |
| 53 | \( 1 - 9.39T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 6.61T + 61T^{2} \) |
| 67 | \( 1 + 3.48T + 67T^{2} \) |
| 71 | \( 1 + 5.18T + 71T^{2} \) |
| 73 | \( 1 + 1.83T + 73T^{2} \) |
| 79 | \( 1 - 2.95T + 79T^{2} \) |
| 83 | \( 1 + 8.37T + 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−7.33495417217510428354838460041, −6.90006444619394038944575127124, −6.35888391175809472211848668960, −5.51595124610418683896476286475, −5.23351995615013903815024332888, −4.18235110208384910377742034114, −4.09835082402861837956307381079, −2.68237276578320053557581459210, −1.63083676721619656295848122034, −0.40958876409728795275964751965,
0.40958876409728795275964751965, 1.63083676721619656295848122034, 2.68237276578320053557581459210, 4.09835082402861837956307381079, 4.18235110208384910377742034114, 5.23351995615013903815024332888, 5.51595124610418683896476286475, 6.35888391175809472211848668960, 6.90006444619394038944575127124, 7.33495417217510428354838460041