| L(s) = 1 | + (−51.7 + 37.6i)2-s + (−630. − 1.94e3i)3-s + (1.26e3 − 3.89e3i)4-s + (5.29e4 + 3.84e4i)5-s + (1.05e5 + 7.67e4i)6-s + (8.61e4 − 2.65e5i)7-s + (8.10e4 + 2.49e5i)8-s + (−2.07e6 + 1.50e6i)9-s − 4.19e6·10-s + (5.90e5 + 5.84e6i)11-s − 8.35e6·12-s + (−1.33e7 + 9.68e6i)13-s + (5.51e6 + 1.69e7i)14-s + (4.12e7 − 1.27e8i)15-s + (−1.35e7 − 9.86e6i)16-s + (3.24e7 + 2.35e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.499 − 1.53i)3-s + (0.154 − 0.475i)4-s + (1.51 + 1.10i)5-s + (0.924 + 0.671i)6-s + (0.276 − 0.852i)7-s + (0.109 + 0.336i)8-s + (−1.30 + 0.946i)9-s − 1.32·10-s + (0.100 + 0.994i)11-s − 0.807·12-s + (−0.766 + 0.556i)13-s + (0.195 + 0.602i)14-s + (0.935 − 2.88i)15-s + (−0.202 − 0.146i)16-s + (0.326 + 0.237i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.520303528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.520303528\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (51.7 - 37.6i)T \) |
| 11 | \( 1 + (-5.90e5 - 5.84e6i)T \) |
| good | 3 | \( 1 + (630. + 1.94e3i)T + (-1.28e6 + 9.37e5i)T^{2} \) |
| 5 | \( 1 + (-5.29e4 - 3.84e4i)T + (3.77e8 + 1.16e9i)T^{2} \) |
| 7 | \( 1 + (-8.61e4 + 2.65e5i)T + (-7.83e10 - 5.69e10i)T^{2} \) |
| 13 | \( 1 + (1.33e7 - 9.68e6i)T + (9.35e13 - 2.88e14i)T^{2} \) |
| 17 | \( 1 + (-3.24e7 - 2.35e7i)T + (3.06e15 + 9.41e15i)T^{2} \) |
| 19 | \( 1 + (-8.74e7 - 2.69e8i)T + (-3.40e16 + 2.47e16i)T^{2} \) |
| 23 | \( 1 - 6.47e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (2.55e8 - 7.84e8i)T + (-8.30e18 - 6.03e18i)T^{2} \) |
| 31 | \( 1 + (6.31e8 - 4.58e8i)T + (7.54e18 - 2.32e19i)T^{2} \) |
| 37 | \( 1 + (-5.58e8 + 1.71e9i)T + (-1.97e20 - 1.43e20i)T^{2} \) |
| 41 | \( 1 + (1.43e10 + 4.40e10i)T + (-7.48e20 + 5.43e20i)T^{2} \) |
| 43 | \( 1 - 4.84e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-3.83e10 - 1.18e11i)T + (-4.41e21 + 3.20e21i)T^{2} \) |
| 53 | \( 1 + (-1.42e11 + 1.03e11i)T + (8.04e21 - 2.47e22i)T^{2} \) |
| 59 | \( 1 + (-1.12e11 + 3.44e11i)T + (-8.49e22 - 6.17e22i)T^{2} \) |
| 61 | \( 1 + (2.61e11 + 1.90e11i)T + (5.00e22 + 1.53e23i)T^{2} \) |
| 67 | \( 1 + 5.47e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-1.34e12 - 9.74e11i)T + (3.60e23 + 1.10e24i)T^{2} \) |
| 73 | \( 1 + (3.63e11 - 1.11e12i)T + (-1.35e24 - 9.82e23i)T^{2} \) |
| 79 | \( 1 + (-7.15e10 + 5.19e10i)T + (1.44e24 - 4.43e24i)T^{2} \) |
| 83 | \( 1 + (-9.12e11 - 6.63e11i)T + (2.74e24 + 8.43e24i)T^{2} \) |
| 89 | \( 1 + 2.15e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (-3.56e12 + 2.58e12i)T + (2.07e25 - 6.40e25i)T^{2} \) |
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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
−14.51984106326600818165145690420, −13.90612841895551867314098466630, −12.50078350610274017069513923647, −10.82977012061334868518083053437, −9.714475420678177826182507168462, −7.41666825283406454184797369638, −6.87587413094582848207523526124, −5.65771738432405797448559351082, −2.17442031452656890360421561723, −1.28339061594208124651510278447,
0.74233964597121141309873203530, 2.69586665285184059659014315715, 4.90327978805840631352181960473, 5.68819219269623268984759153032, 8.841027486747929116807095775094, 9.409811630782087248587022405718, 10.52140281449753472149030237677, 11.86975843789945818303358659925, 13.40932936202093315076635821854, 15.15841655807070018077001047910
