L(s) = 1 | + 9i·3-s − 46.1i·5-s + 59.9·7-s − 81·9-s − 400. i·11-s + 351. i·13-s + 415.·15-s + 1.66e3·17-s + 564. i·19-s + 539. i·21-s − 3.83e3·23-s + 994.·25-s − 729i·27-s − 5.94e3i·29-s − 2.67e3·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.825i·5-s + 0.462·7-s − 0.333·9-s − 0.996i·11-s + 0.577i·13-s + 0.476·15-s + 1.39·17-s + 0.358i·19-s + 0.266i·21-s − 1.51·23-s + 0.318·25-s − 0.192i·27-s − 1.31i·29-s − 0.500·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.598026246\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.598026246\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 9iT \) |
good | 5 | \( 1 + 46.1iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 59.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 400. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 351. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.66e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 564. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.83e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.94e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 2.67e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 771. iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 5.11e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.51e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 8.13e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.10e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.21e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 3.54e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.13e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.00e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.23e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.51e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.58e5T + 8.58e9T^{2} \) |
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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
−10.16670637827594777107596396354, −9.426219292694398787103806650747, −8.395114725981058564849325378509, −7.85080098471815486650521102793, −6.19302623326527930285964853595, −5.37523273070430319040007785375, −4.36256015772214245588481108435, −3.35931519391025746530462114422, −1.71972848197837880656052869215, −0.41165996980457245095582716080,
1.26255104526552920727659174361, 2.43265586306210158128717190850, 3.57344095878073701454716451814, 5.02750345301009567468539689860, 6.05087586728063504510733239123, 7.20460003829707803410281333275, 7.66826327042265428974878493988, 8.802741401080140718848423878701, 10.08984366834395388596213221022, 10.61356792831596658195798746336