| 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
−16.07247691963056998619413864202, −15.64034098328326448944549825796, −13.74174834547093114223503895340, −12.49031426092037699941458992059, −11.44852038859294762534800465032, −10.14097445219221923098528760050, −8.874229566981841457696487690154, −5.77242469759437413590588355245, −4.32399780164893432338349480346, −3.50112482218346242213603834839,
0.37707371193806150927389542891, 3.22661268158982144642157831420, 6.02529995925586523197312667166, 6.52814932910201296376835001095, 7.79270226492289278859767467443, 10.53315757331490568943015138541, 12.22497425565343564877603945159, 13.13574973241795054372633592642, 13.88362773625367682222342609185, 15.48911273378807731103267896427
