L(s) = 1 | − 5.35·2-s − 1.66·3-s + 20.6·4-s + 5.96·5-s + 8.89·6-s + 27.8·7-s − 67.6·8-s − 24.2·9-s − 31.9·10-s − 18.6·11-s − 34.3·12-s − 42.5·13-s − 148.·14-s − 9.92·15-s + 197.·16-s + 129.·18-s + 31.3·19-s + 123.·20-s − 46.2·21-s + 99.7·22-s − 60.3·23-s + 112.·24-s − 89.3·25-s + 227.·26-s + 85.1·27-s + 574.·28-s − 117.·29-s + ⋯ |
L(s) = 1 | − 1.89·2-s − 0.319·3-s + 2.58·4-s + 0.533·5-s + 0.605·6-s + 1.50·7-s − 2.99·8-s − 0.897·9-s − 1.01·10-s − 0.510·11-s − 0.825·12-s − 0.907·13-s − 2.84·14-s − 0.170·15-s + 3.07·16-s + 1.69·18-s + 0.378·19-s + 1.37·20-s − 0.480·21-s + 0.966·22-s − 0.546·23-s + 0.956·24-s − 0.714·25-s + 1.71·26-s + 0.607·27-s + 3.87·28-s − 0.753·29-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 | \( 1 + 5.35T + 8T^{2} \) |
| 3 | \( 1 + 1.66T + 27T^{2} \) |
| 5 | \( 1 - 5.96T + 125T^{2} \) |
| 7 | \( 1 - 27.8T + 343T^{2} \) |
| 11 | \( 1 + 18.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 42.5T + 2.19e3T^{2} \) |
| 19 | \( 1 - 31.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 60.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 117.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 228.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 99.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 270.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 108.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 294.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 62.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 799.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 645.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.14e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 550.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 253.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 717.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.59e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 255.T + 9.12e5T^{2} \) |
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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
−10.65617097282784599749517427600, −9.975544376638596357851506874852, −8.937523537200365733646775825100, −8.091507842984438318797154214346, −7.47545445696274404815493409644, −6.13163564914287770027752239680, −5.11934844033820805403519251771, −2.60116633966459032028945946288, −1.57884593107119191998622730356, 0,
1.57884593107119191998622730356, 2.60116633966459032028945946288, 5.11934844033820805403519251771, 6.13163564914287770027752239680, 7.47545445696274404815493409644, 8.091507842984438318797154214346, 8.937523537200365733646775825100, 9.975544376638596357851506874852, 10.65617097282784599749517427600