| L(s) = 1 | + 3.25·2-s + 6.51·3-s + 2.61·4-s − 19.1·5-s + 21.2·6-s − 22.0·7-s − 17.5·8-s + 15.4·9-s − 62.2·10-s + 8.25·11-s + 17.0·12-s − 0.398·13-s − 71.9·14-s − 124.·15-s − 78.0·16-s + 50.4·18-s + 103.·19-s − 49.8·20-s − 144.·21-s + 26.9·22-s − 138.·23-s − 114.·24-s + 240.·25-s − 1.29·26-s − 75.0·27-s − 57.7·28-s − 222.·29-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 1.25·3-s + 0.326·4-s − 1.70·5-s + 1.44·6-s − 1.19·7-s − 0.775·8-s + 0.573·9-s − 1.96·10-s + 0.226·11-s + 0.409·12-s − 0.00849·13-s − 1.37·14-s − 2.14·15-s − 1.21·16-s + 0.660·18-s + 1.24·19-s − 0.557·20-s − 1.49·21-s + 0.260·22-s − 1.25·23-s − 0.973·24-s + 1.92·25-s − 0.00978·26-s − 0.535·27-s − 0.389·28-s − 1.42·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 - 3.25T + 8T^{2} \) |
| 3 | \( 1 - 6.51T + 27T^{2} \) |
| 5 | \( 1 + 19.1T + 125T^{2} \) |
| 7 | \( 1 + 22.0T + 343T^{2} \) |
| 11 | \( 1 - 8.25T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.398T + 2.19e3T^{2} \) |
| 19 | \( 1 - 103.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 222.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 49.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 61.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 387.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 20.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 44.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 59.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 238.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 595.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 408.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 22.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 682.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 312.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 904.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.00e3T + 9.12e5T^{2} \) |
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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
−11.34131495302142249924614762567, −9.692412405296191286707259317252, −8.939867602686518087048767475548, −7.907298949317358968981149883233, −7.07764531794226722262973427930, −5.69404882000246178489093933655, −4.08995932751701900968821899048, −3.63426010047432223921482666656, −2.82323790110148593813724749730, 0,
2.82323790110148593813724749730, 3.63426010047432223921482666656, 4.08995932751701900968821899048, 5.69404882000246178489093933655, 7.07764531794226722262973427930, 7.907298949317358968981149883233, 8.939867602686518087048767475548, 9.692412405296191286707259317252, 11.34131495302142249924614762567
