| L(s) = 1 | + (14.3 − 10.4i)2-s + (13.2 − 40.8i)3-s + (57.5 − 177. i)4-s + (−267. − 80.6i)5-s + (−235. − 723. i)6-s + 6.21·7-s + (−319. − 982. i)8-s + (277. + 201. i)9-s + (−4.67e3 + 1.63e3i)10-s + (4.29e3 − 3.11e3i)11-s + (−6.46e3 − 4.70e3i)12-s + (648. + 471. i)13-s + (89.1 − 64.7i)14-s + (−6.84e3 + 9.85e3i)15-s + (4.47e3 + 3.25e3i)16-s + (−1.87e3 − 5.77e3i)17-s + ⋯ |
| L(s) = 1 | + (1.26 − 0.920i)2-s + (0.283 − 0.873i)3-s + (0.449 − 1.38i)4-s + (−0.957 − 0.288i)5-s + (−0.444 − 1.36i)6-s + 0.00684·7-s + (−0.220 − 0.678i)8-s + (0.127 + 0.0923i)9-s + (−1.47 + 0.515i)10-s + (0.972 − 0.706i)11-s + (−1.08 − 0.785i)12-s + (0.0819 + 0.0595i)13-s + (0.00868 − 0.00630i)14-s + (−0.523 + 0.754i)15-s + (0.273 + 0.198i)16-s + (−0.0926 − 0.285i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.20400 - 2.86131i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20400 - 2.86131i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (267. + 80.6i)T \) |
| good | 2 | \( 1 + (-14.3 + 10.4i)T + (39.5 - 121. i)T^{2} \) |
| 3 | \( 1 + (-13.2 + 40.8i)T + (-1.76e3 - 1.28e3i)T^{2} \) |
| 7 | \( 1 - 6.21T + 8.23e5T^{2} \) |
| 11 | \( 1 + (-4.29e3 + 3.11e3i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (-648. - 471. i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (1.87e3 + 5.77e3i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (9.35e3 + 2.87e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (-1.76e4 + 1.27e4i)T + (1.05e9 - 3.23e9i)T^{2} \) |
| 29 | \( 1 + (7.93e4 - 2.44e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-5.48e4 - 1.68e5i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-3.05e5 - 2.21e5i)T + (2.93e10 + 9.02e10i)T^{2} \) |
| 41 | \( 1 + (7.39e4 + 5.37e4i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + 8.06e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-2.47e5 + 7.62e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (3.50e5 - 1.07e6i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (-7.22e5 - 5.24e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (1.63e6 - 1.19e6i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (-9.58e4 - 2.94e5i)T + (-4.90e12 + 3.56e12i)T^{2} \) |
| 71 | \( 1 + (-1.46e6 + 4.49e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (5.90e5 - 4.29e5i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (1.02e6 - 3.16e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-9.71e5 - 2.98e6i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + (5.56e6 - 4.04e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (-5.06e6 + 1.55e7i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−15.03456607838189475505375436271, −13.85751111762788840510281096154, −12.90199010833133044283231369632, −11.95411449668797330127148224138, −10.96417738755488336097144475313, −8.581235990358388593844591567715, −6.84701061844676868711717125947, −4.72743065466283624409408312915, −3.20516769694937936844720423919, −1.28075250301645057509514696202,
3.69556526986423021993239205529, 4.43995805756042850376680148810, 6.41435311091424216245190883835, 7.84005383914420392992994767479, 9.774698867242117553487381181761, 11.64029525174610994167030597730, 12.88293682411989502212962060143, 14.55847094966563602409102496384, 15.03938419898664382427490657673, 15.91960456241077523437467970757
