| L(s) = 1 | + (−32 + 55.4i)2-s + (−855. − 1.48e3i)3-s + (−2.04e3 − 3.54e3i)4-s + (1.25e4 − 2.17e4i)5-s + 1.09e5·6-s + (4.88e4 − 3.07e5i)7-s + 2.62e5·8-s + (−6.67e5 + 1.15e6i)9-s + (8.02e5 + 1.39e6i)10-s + (8.50e5 + 1.47e6i)11-s + (−3.50e6 + 6.07e6i)12-s − 9.03e6·13-s + (1.54e7 + 1.25e7i)14-s − 4.29e7·15-s + (−8.38e6 + 1.45e7i)16-s + (−9.01e7 − 1.56e8i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.677 − 1.17i)3-s + (−0.249 − 0.433i)4-s + (0.359 − 0.621i)5-s + 0.958·6-s + (0.156 − 0.987i)7-s + 0.353·8-s + (−0.418 + 0.724i)9-s + (0.253 + 0.439i)10-s + (0.144 + 0.250i)11-s + (−0.338 + 0.586i)12-s − 0.518·13-s + (0.549 + 0.445i)14-s − 0.973·15-s + (−0.125 + 0.216i)16-s + (−0.905 − 1.56i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0939i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.0261589 + 0.555591i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0261589 + 0.555591i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (32 - 55.4i)T \) |
| 7 | \( 1 + (-4.88e4 + 3.07e5i)T \) |
| good | 3 | \( 1 + (855. + 1.48e3i)T + (-7.97e5 + 1.38e6i)T^{2} \) |
| 5 | \( 1 + (-1.25e4 + 2.17e4i)T + (-6.10e8 - 1.05e9i)T^{2} \) |
| 11 | \( 1 + (-8.50e5 - 1.47e6i)T + (-1.72e13 + 2.98e13i)T^{2} \) |
| 13 | \( 1 + 9.03e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + (9.01e7 + 1.56e8i)T + (-4.95e15 + 8.57e15i)T^{2} \) |
| 19 | \( 1 + (1.70e7 - 2.94e7i)T + (-2.10e16 - 3.64e16i)T^{2} \) |
| 23 | \( 1 + (5.27e8 - 9.12e8i)T + (-2.52e17 - 4.36e17i)T^{2} \) |
| 29 | \( 1 - 2.18e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + (-3.02e8 - 5.24e8i)T + (-1.22e19 + 2.11e19i)T^{2} \) |
| 37 | \( 1 + (8.94e9 - 1.54e10i)T + (-1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 - 4.48e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 3.24e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + (3.67e10 - 6.37e10i)T + (-2.73e21 - 4.72e21i)T^{2} \) |
| 53 | \( 1 + (1.45e11 + 2.51e11i)T + (-1.30e22 + 2.25e22i)T^{2} \) |
| 59 | \( 1 + (1.55e11 + 2.69e11i)T + (-5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (-2.79e11 + 4.84e11i)T + (-8.09e22 - 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-4.47e11 - 7.74e11i)T + (-2.74e23 + 4.74e23i)T^{2} \) |
| 71 | \( 1 + 6.75e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + (9.03e11 + 1.56e12i)T + (-8.35e23 + 1.44e24i)T^{2} \) |
| 79 | \( 1 + (-1.57e12 + 2.72e12i)T + (-2.33e24 - 4.04e24i)T^{2} \) |
| 83 | \( 1 + 3.34e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (3.34e12 - 5.79e12i)T + (-1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 - 1.01e13T + 6.73e25T^{2} \) |
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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
−16.06701244382730610634691652102, −14.01994895253216205814013575776, −13.00829694833716492430585396675, −11.49291918581611451783976080370, −9.589626736158788805097373249906, −7.66339074797392185411976802537, −6.63587916801606017246658549903, −4.96156213248973379642376791800, −1.48837903889266847969132186989, −0.27951950575615446141902666651,
2.38626363067284149308425982890, 4.33351481157201291120707662118, 6.05699737678555855878012955656, 8.717662488735412062520138622974, 10.16716558914762775716981859700, 11.01357970857286233864864743460, 12.43366484530904026415045783156, 14.56408175583542491290067082041, 15.81095771238611173442323679024, 17.17320211444410499545341227728
