L(s) = 1 | + (7.06 − 14.3i)2-s + 134.·3-s + (−156. − 202. i)4-s + (−416. − 466. i)5-s + (953. − 1.93e3i)6-s + 1.86e3·7-s + (−4.01e3 + 810. i)8-s + 1.16e4·9-s + (−9.63e3 + 2.68e3i)10-s + 5.27e3i·11-s + (−2.10e4 − 2.73e4i)12-s − 3.33e4i·13-s + (1.31e4 − 2.67e4i)14-s + (−5.61e4 − 6.29e4i)15-s + (−1.67e4 + 6.33e4i)16-s + 1.42e5i·17-s + ⋯ |
L(s) = 1 | + (0.441 − 0.897i)2-s + 1.66·3-s + (−0.610 − 0.792i)4-s + (−0.665 − 0.745i)5-s + (0.735 − 1.49i)6-s + 0.776·7-s + (−0.980 + 0.197i)8-s + 1.77·9-s + (−0.963 + 0.268i)10-s + 0.360i·11-s + (−1.01 − 1.32i)12-s − 1.16i·13-s + (0.342 − 0.696i)14-s + (−1.10 − 1.24i)15-s + (−0.255 + 0.966i)16-s + 1.70i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.95308 - 2.35402i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95308 - 2.35402i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-7.06 + 14.3i)T \) |
| 5 | \( 1 + (416. + 466. i)T \) |
good | 3 | \( 1 - 134.T + 6.56e3T^{2} \) |
| 7 | \( 1 - 1.86e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 5.27e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 3.33e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 1.42e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 7.03e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 2.55e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 5.34e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.24e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 1.69e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 2.36e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 4.03e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 1.65e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 9.81e5iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 7.47e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.41e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.75e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 3.57e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 7.26e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 3.72e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 2.09e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 2.52e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 1.35e6iT - 7.83e15T^{2} \) |
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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
−15.34018281839576260255709643211, −14.74819519658190092514107410676, −13.32913354434942368293188598810, −12.42827466551313533383041404954, −10.58074698738177080243651730190, −8.924221294685181728251241945913, −8.040368710291411283809150438742, −4.68754746942266332936594345090, −3.25119582872249441225565484376, −1.47018518441270629900283763672,
2.85364170137941281454226843962, 4.32196929892531219860602942141, 7.07375358545588526634389257051, 8.055548641951582219923312966159, 9.268074989044433903294945924323, 11.67902183523030821216911276182, 13.63585595426389423199615098776, 14.34124808499945407159522591946, 15.13625000563123761055406270416, 16.31422760799093852009655525076