| L(s) = 1 | − 1.90·3-s + 18.0·5-s − 0.923·7-s − 23.3·9-s + 49.9·11-s − 78.4·13-s − 34.2·15-s + 81.5·17-s − 103.·19-s + 1.75·21-s − 23·23-s + 200.·25-s + 95.7·27-s − 153.·29-s + 59.6·31-s − 95.0·33-s − 16.6·35-s − 138.·37-s + 149.·39-s + 28.5·41-s − 359.·43-s − 421.·45-s − 554.·47-s − 342.·49-s − 155.·51-s − 362.·53-s + 901.·55-s + ⋯ |
| L(s) = 1 | − 0.365·3-s + 1.61·5-s − 0.0498·7-s − 0.866·9-s + 1.37·11-s − 1.67·13-s − 0.590·15-s + 1.16·17-s − 1.25·19-s + 0.0182·21-s − 0.208·23-s + 1.60·25-s + 0.682·27-s − 0.984·29-s + 0.345·31-s − 0.501·33-s − 0.0804·35-s − 0.616·37-s + 0.612·39-s + 0.108·41-s − 1.27·43-s − 1.39·45-s − 1.72·47-s − 0.997·49-s − 0.425·51-s − 0.939·53-s + 2.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 1.90T + 27T^{2} \) |
| 5 | \( 1 - 18.0T + 125T^{2} \) |
| 7 | \( 1 + 0.923T + 343T^{2} \) |
| 11 | \( 1 - 49.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 78.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 81.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 103.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 153.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 59.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 138.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 28.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 359.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 554.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 362.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 736.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 643.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 503.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 760.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 674.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 538.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 248.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.936430820073506236060026919231, −8.011283121166177593240825140481, −6.73975840882208789785603342441, −6.32537844047674372901094529973, −5.44130859703355815378954090392, −4.83475432525661149196394833623, −3.42051288955039291976496039839, −2.32103249233261865797741625833, −1.50400768369469858657927276193, 0,
1.50400768369469858657927276193, 2.32103249233261865797741625833, 3.42051288955039291976496039839, 4.83475432525661149196394833623, 5.44130859703355815378954090392, 6.32537844047674372901094529973, 6.73975840882208789785603342441, 8.011283121166177593240825140481, 8.936430820073506236060026919231
