| L(s) = 1 | + (10.7 + 7.83i)2-s + (−25.0 − 76.9i)3-s + (15.2 + 47.0i)4-s + (−143. + 240. i)5-s + (333. − 1.02e3i)6-s − 1.42e3·7-s + (323. − 994. i)8-s + (−3.53e3 + 2.56e3i)9-s + (−3.42e3 + 1.46e3i)10-s + (−1.78e3 − 1.29e3i)11-s + (3.24e3 − 2.35e3i)12-s + (9.27e3 − 6.73e3i)13-s + (−1.53e4 − 1.11e4i)14-s + (2.20e4 + 5.00e3i)15-s + (1.63e4 − 1.19e4i)16-s + (982. − 3.02e3i)17-s + ⋯ |
| L(s) = 1 | + (0.952 + 0.692i)2-s + (−0.534 − 1.64i)3-s + (0.119 + 0.367i)4-s + (−0.511 + 0.859i)5-s + (0.629 − 1.93i)6-s − 1.56·7-s + (0.223 − 0.686i)8-s + (−1.61 + 1.17i)9-s + (−1.08 + 0.464i)10-s + (−0.404 − 0.293i)11-s + (0.541 − 0.393i)12-s + (1.17 − 0.850i)13-s + (−1.49 − 1.08i)14-s + (1.68 + 0.382i)15-s + (1.00 − 0.727i)16-s + (0.0484 − 0.149i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.317326 - 0.889941i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.317326 - 0.889941i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (143. - 240. i)T \) |
| good | 2 | \( 1 + (-10.7 - 7.83i)T + (39.5 + 121. i)T^{2} \) |
| 3 | \( 1 + (25.0 + 76.9i)T + (-1.76e3 + 1.28e3i)T^{2} \) |
| 7 | \( 1 + 1.42e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + (1.78e3 + 1.29e3i)T + (6.02e6 + 1.85e7i)T^{2} \) |
| 13 | \( 1 + (-9.27e3 + 6.73e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-982. + 3.02e3i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (1.33e3 - 4.09e3i)T + (-7.23e8 - 5.25e8i)T^{2} \) |
| 23 | \( 1 + (5.42e4 + 3.94e4i)T + (1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-3.40e4 - 1.04e5i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-4.72e4 + 1.45e5i)T + (-2.22e10 - 1.61e10i)T^{2} \) |
| 37 | \( 1 + (1.02e5 - 7.47e4i)T + (2.93e10 - 9.02e10i)T^{2} \) |
| 41 | \( 1 + (4.96e5 - 3.60e5i)T + (6.01e10 - 1.85e11i)T^{2} \) |
| 43 | \( 1 - 6.87e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (8.41e4 + 2.58e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-7.26e3 - 2.23e4i)T + (-9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (1.12e6 - 8.15e5i)T + (7.69e11 - 2.36e12i)T^{2} \) |
| 61 | \( 1 + (1.55e6 + 1.12e6i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 + (-9.66e5 + 2.97e6i)T + (-4.90e12 - 3.56e12i)T^{2} \) |
| 71 | \( 1 + (1.41e6 + 4.36e6i)T + (-7.35e12 + 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-8.09e4 - 5.88e4i)T + (3.41e12 + 1.05e13i)T^{2} \) |
| 79 | \( 1 + (2.28e5 + 7.03e5i)T + (-1.55e13 + 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-3.86e5 + 1.18e6i)T + (-2.19e13 - 1.59e13i)T^{2} \) |
| 89 | \( 1 + (-1.52e6 - 1.10e6i)T + (1.36e13 + 4.20e13i)T^{2} \) |
| 97 | \( 1 + (-1.65e6 - 5.09e6i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−15.48911273378807731103267896427, −13.88362773625367682222342609185, −13.13574973241795054372633592642, −12.22497425565343564877603945159, −10.53315757331490568943015138541, −7.79270226492289278859767467443, −6.52814932910201296376835001095, −6.02529995925586523197312667166, −3.22661268158982144642157831420, −0.37707371193806150927389542891,
3.50112482218346242213603834839, 4.32399780164893432338349480346, 5.77242469759437413590588355245, 8.874229566981841457696487690154, 10.14097445219221923098528760050, 11.44852038859294762534800465032, 12.49031426092037699941458992059, 13.74174834547093114223503895340, 15.64034098328326448944549825796, 16.07247691963056998619413864202
