L(s) = 1 | + 2-s − 1.56·3-s + 4-s − 1.56·6-s − 1.56·7-s + 8-s − 0.561·9-s + 4·11-s − 1.56·12-s + 6.68·13-s − 1.56·14-s + 16-s − 7.56·17-s − 0.561·18-s − 19-s + 2.43·21-s + 4·22-s + 4.68·23-s − 1.56·24-s + 6.68·26-s + 5.56·27-s − 1.56·28-s + 6.68·29-s + 3.12·31-s + 32-s − 6.24·33-s − 7.56·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.901·3-s + 0.5·4-s − 0.637·6-s − 0.590·7-s + 0.353·8-s − 0.187·9-s + 1.20·11-s − 0.450·12-s + 1.85·13-s − 0.417·14-s + 0.250·16-s − 1.83·17-s − 0.132·18-s − 0.229·19-s + 0.532·21-s + 0.852·22-s + 0.976·23-s − 0.318·24-s + 1.31·26-s + 1.07·27-s − 0.295·28-s + 1.24·29-s + 0.560·31-s + 0.176·32-s − 1.08·33-s − 1.29·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.824081225\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.824081225\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 6.68T + 13T^{2} \) |
| 17 | \( 1 + 7.56T + 17T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 0.438T + 53T^{2} \) |
| 59 | \( 1 + 1.56T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 + 1.56T + 67T^{2} \) |
| 71 | \( 1 + 6.24T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−10.40509723935053206255780021760, −9.030349635085689639443277468277, −8.607425023989674458271902987602, −7.00515204683514271804791357069, −6.25170554024009670678048785667, −6.06325101073125175662248675889, −4.67093250850662199388310576228, −3.94670328982411401578264404479, −2.76636362403408315057351541959, −1.06121728191310280520996406203,
1.06121728191310280520996406203, 2.76636362403408315057351541959, 3.94670328982411401578264404479, 4.67093250850662199388310576228, 6.06325101073125175662248675889, 6.25170554024009670678048785667, 7.00515204683514271804791357069, 8.607425023989674458271902987602, 9.030349635085689639443277468277, 10.40509723935053206255780021760