| L(s) = 1 | − 4.58·2-s + 12.9·4-s + 17.0·5-s − 20.9·7-s − 22.8·8-s − 77.9·10-s + 48.1·11-s + 29.4·13-s + 96.1·14-s + 0.636·16-s − 108.·17-s − 111.·19-s + 220.·20-s − 220.·22-s + 14.3·23-s + 164.·25-s − 134.·26-s − 272.·28-s − 292.·29-s − 216.·31-s + 179.·32-s + 495.·34-s − 357.·35-s − 168.·37-s + 509.·38-s − 388.·40-s + 342.·41-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.62·4-s + 1.52·5-s − 1.13·7-s − 1.00·8-s − 2.46·10-s + 1.32·11-s + 0.628·13-s + 1.83·14-s + 0.00994·16-s − 1.54·17-s − 1.34·19-s + 2.46·20-s − 2.13·22-s + 0.130·23-s + 1.31·25-s − 1.01·26-s − 1.83·28-s − 1.87·29-s − 1.25·31-s + 0.991·32-s + 2.50·34-s − 1.72·35-s − 0.748·37-s + 2.17·38-s − 1.53·40-s + 1.30·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 59 | \( 1 + 59T \) |
| good | 2 | \( 1 + 4.58T + 8T^{2} \) |
| 5 | \( 1 - 17.0T + 125T^{2} \) |
| 7 | \( 1 + 20.9T + 343T^{2} \) |
| 11 | \( 1 - 48.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 29.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 14.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 292.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 216.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 168.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 342.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 325.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 315.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 256.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 749.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 664.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 207.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 45.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 813.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 19.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 636.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 235.T + 9.12e5T^{2} \) |
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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
−9.602483749967519581901115381305, −9.172736679767072777717269801546, −8.811774018714152198190556193597, −7.21607156201943877598702265951, −6.44364944214156394350763997716, −5.95434506075618742336872389846, −4.01401674117577131292552476067, −2.34547401846831520331106287829, −1.53474343808844129337739199063, 0,
1.53474343808844129337739199063, 2.34547401846831520331106287829, 4.01401674117577131292552476067, 5.95434506075618742336872389846, 6.44364944214156394350763997716, 7.21607156201943877598702265951, 8.811774018714152198190556193597, 9.172736679767072777717269801546, 9.602483749967519581901115381305
