| L(s) = 1 | − 2·2-s − 2.73·3-s + 4·4-s + 20.6·5-s + 5.47·6-s − 27.1·7-s − 8·8-s − 19.5·9-s − 41.3·10-s + 27.4·11-s − 10.9·12-s + 64.9·13-s + 54.2·14-s − 56.5·15-s + 16·16-s − 38.8·17-s + 39.0·18-s + 57.8·19-s + 82.6·20-s + 74.1·21-s − 54.9·22-s + 80.7·23-s + 21.8·24-s + 302.·25-s − 129.·26-s + 127.·27-s − 108.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.526·3-s + 0.5·4-s + 1.84·5-s + 0.372·6-s − 1.46·7-s − 0.353·8-s − 0.722·9-s − 1.30·10-s + 0.753·11-s − 0.263·12-s + 1.38·13-s + 1.03·14-s − 0.973·15-s + 0.250·16-s − 0.554·17-s + 0.510·18-s + 0.698·19-s + 0.924·20-s + 0.770·21-s − 0.532·22-s + 0.732·23-s + 0.186·24-s + 2.41·25-s − 0.979·26-s + 0.907·27-s − 0.731·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1922 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1922 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.588997003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.588997003\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + 2.73T + 27T^{2} \) |
| 5 | \( 1 - 20.6T + 125T^{2} \) |
| 7 | \( 1 + 27.1T + 343T^{2} \) |
| 11 | \( 1 - 27.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 38.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 57.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 80.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 198.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 306.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 323.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 163.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 71.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 706.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 653.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 139.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 372.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 832.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 438.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 350.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 255.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 555.T + 9.12e5T^{2} \) |
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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
−9.194290038170652398494709442913, −8.381663996928330490045888264250, −6.84405901887744931201961901344, −6.44406589410483441381275909461, −5.93537778975929162287226853544, −5.23966377760300977544821744743, −3.55875406171885341635751257891, −2.76060714502974006013835096373, −1.64287178888254869162142396504, −0.68707690396649883597675310660,
0.68707690396649883597675310660, 1.64287178888254869162142396504, 2.76060714502974006013835096373, 3.55875406171885341635751257891, 5.23966377760300977544821744743, 5.93537778975929162287226853544, 6.44406589410483441381275909461, 6.84405901887744931201961901344, 8.381663996928330490045888264250, 9.194290038170652398494709442913
