| L(s) = 1 | − 3·3-s + 1.26·5-s + 28.5·7-s + 9·9-s + 26.2·11-s + 13·13-s − 3.79·15-s − 37.8·17-s + 119.·19-s − 85.6·21-s − 40.5·23-s − 123.·25-s − 27·27-s − 147.·29-s − 328.·31-s − 78.6·33-s + 36.1·35-s − 32.1·37-s − 39·39-s − 456.·41-s − 109.·43-s + 11.3·45-s − 347.·47-s + 471.·49-s + 113.·51-s + 356.·53-s + 33.1·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.113·5-s + 1.54·7-s + 0.333·9-s + 0.718·11-s + 0.277·13-s − 0.0653·15-s − 0.540·17-s + 1.44·19-s − 0.889·21-s − 0.367·23-s − 0.987·25-s − 0.192·27-s − 0.942·29-s − 1.90·31-s − 0.415·33-s + 0.174·35-s − 0.142·37-s − 0.160·39-s − 1.73·41-s − 0.388·43-s + 0.0377·45-s − 1.07·47-s + 1.37·49-s + 0.311·51-s + 0.924·53-s + 0.0813·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 1.26T + 125T^{2} \) |
| 7 | \( 1 - 28.5T + 343T^{2} \) |
| 11 | \( 1 - 26.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 37.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 40.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 147.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 328.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 32.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 456.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 109.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 347.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 356.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 193.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 473.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 377.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 674.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 260.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 279.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 195.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 183.T + 9.12e5T^{2} \) |
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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
−8.077124620166812307129675997127, −7.46531720596180342926648066251, −6.68177473618471898324273487116, −5.62271379638761431471124268346, −5.20395736420620344591633071549, −4.26236364008592961542941841081, −3.46403079405926744598967454394, −1.85063045205687387096191855646, −1.43617772089389835892523401769, 0,
1.43617772089389835892523401769, 1.85063045205687387096191855646, 3.46403079405926744598967454394, 4.26236364008592961542941841081, 5.20395736420620344591633071549, 5.62271379638761431471124268346, 6.68177473618471898324273487116, 7.46531720596180342926648066251, 8.077124620166812307129675997127
