L(s) = 1 | − 15.1i·2-s − 101.·4-s + 198. i·7-s − 407. i·8-s + 5.26e3·11-s − 1.21e3i·13-s + 3.01e3·14-s − 1.91e4·16-s + 3.45e4i·17-s − 1.86e4·19-s − 7.97e4i·22-s − 3.33e4i·23-s − 1.83e4·26-s − 2.01e4i·28-s − 1.78e5·29-s + ⋯ |
L(s) = 1 | − 1.33i·2-s − 0.789·4-s + 0.219i·7-s − 0.281i·8-s + 1.19·11-s − 0.153i·13-s + 0.293·14-s − 1.16·16-s + 1.70i·17-s − 0.622·19-s − 1.59i·22-s − 0.571i·23-s − 0.204·26-s − 0.173i·28-s − 1.35·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.003339963\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.003339963\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 15.1iT - 128T^{2} \) |
| 7 | \( 1 - 198. iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 5.26e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.21e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 3.45e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 1.86e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.33e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 1.78e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.37e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.82e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 2.93e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.43e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 4.81e4iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 1.66e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.75e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.15e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.29e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 2.71e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.67e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 3.44e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.71e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 3.52e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.44e7iT - 8.07e13T^{2} \) |
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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
−11.02171050549436011934229303042, −10.36382351851152950774791296979, −9.296726560775652327773767097115, −8.500738482843575767565582786495, −6.95790183872105342483868290093, −5.90086933977669158382939758224, −4.23588650533910130567835760838, −3.51610116067441014808590509283, −2.10119271949157578938358413146, −1.26996731575291439509062214494,
0.23572036980082100413677619363, 1.95937923993564415926605233609, 3.71442340291824195646278866801, 4.95488473810286878307662097836, 5.95720125054541284937205080928, 7.00725199200654729804502199138, 7.54207835679177143591827688137, 8.908344785342820601076684442870, 9.448643486472429059592010046414, 11.05471225350237690543740736342