| L(s) = 1 | + (−133. + 96.6i)2-s + (−300. − 925. i)3-s + (5.82e3 − 1.79e4i)4-s + (6.36e3 + 4.62e3i)5-s + (1.29e5 + 9.40e4i)6-s + (−6.24e4 + 1.92e5i)7-s + (5.42e5 + 1.66e6i)8-s + (5.24e5 − 3.80e5i)9-s − 1.29e6·10-s + (3.48e6 − 4.73e6i)11-s − 1.83e7·12-s + (−8.64e6 + 6.27e6i)13-s + (−1.02e7 − 3.15e7i)14-s + (2.36e6 − 7.27e6i)15-s + (−1.08e8 − 7.87e7i)16-s + (−1.08e8 − 7.91e7i)17-s + ⋯ |
| L(s) = 1 | + (−1.47 + 1.06i)2-s + (−0.238 − 0.732i)3-s + (0.711 − 2.18i)4-s + (0.182 + 0.132i)5-s + (1.13 + 0.823i)6-s + (−0.200 + 0.617i)7-s + (0.731 + 2.25i)8-s + (0.328 − 0.238i)9-s − 0.409·10-s + (0.593 − 0.805i)11-s − 1.77·12-s + (−0.496 + 0.360i)13-s + (−0.364 − 1.12i)14-s + (0.0535 − 0.164i)15-s + (−1.61 − 1.17i)16-s + (−1.09 − 0.795i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.00836656 - 0.0585835i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00836656 - 0.0585835i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-3.48e6 + 4.73e6i)T \) |
| good | 2 | \( 1 + (133. - 96.6i)T + (2.53e3 - 7.79e3i)T^{2} \) |
| 3 | \( 1 + (300. + 925. i)T + (-1.28e6 + 9.37e5i)T^{2} \) |
| 5 | \( 1 + (-6.36e3 - 4.62e3i)T + (3.77e8 + 1.16e9i)T^{2} \) |
| 7 | \( 1 + (6.24e4 - 1.92e5i)T + (-7.83e10 - 5.69e10i)T^{2} \) |
| 13 | \( 1 + (8.64e6 - 6.27e6i)T + (9.35e13 - 2.88e14i)T^{2} \) |
| 17 | \( 1 + (1.08e8 + 7.91e7i)T + (3.06e15 + 9.41e15i)T^{2} \) |
| 19 | \( 1 + (-8.65e7 - 2.66e8i)T + (-3.40e16 + 2.47e16i)T^{2} \) |
| 23 | \( 1 + 8.47e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (1.47e9 - 4.52e9i)T + (-8.30e18 - 6.03e18i)T^{2} \) |
| 31 | \( 1 + (7.49e8 - 5.44e8i)T + (7.54e18 - 2.32e19i)T^{2} \) |
| 37 | \( 1 + (9.23e9 - 2.84e10i)T + (-1.97e20 - 1.43e20i)T^{2} \) |
| 41 | \( 1 + (4.47e9 + 1.37e10i)T + (-7.48e20 + 5.43e20i)T^{2} \) |
| 43 | \( 1 + 2.03e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (1.97e10 + 6.07e10i)T + (-4.41e21 + 3.20e21i)T^{2} \) |
| 53 | \( 1 + (5.97e9 - 4.34e9i)T + (8.04e21 - 2.47e22i)T^{2} \) |
| 59 | \( 1 + (8.12e10 - 2.50e11i)T + (-8.49e22 - 6.17e22i)T^{2} \) |
| 61 | \( 1 + (7.83e10 + 5.69e10i)T + (5.00e22 + 1.53e23i)T^{2} \) |
| 67 | \( 1 + 2.03e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (1.61e11 + 1.17e11i)T + (3.60e23 + 1.10e24i)T^{2} \) |
| 73 | \( 1 + (-3.77e10 + 1.16e11i)T + (-1.35e24 - 9.82e23i)T^{2} \) |
| 79 | \( 1 + (3.30e12 - 2.40e12i)T + (1.44e24 - 4.43e24i)T^{2} \) |
| 83 | \( 1 + (1.97e12 + 1.43e12i)T + (2.74e24 + 8.43e24i)T^{2} \) |
| 89 | \( 1 + 7.54e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (-8.32e12 + 6.05e12i)T + (2.07e25 - 6.40e25i)T^{2} \) |
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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
−18.16925353342922602924207749554, −16.77500674678503705448810114442, −15.66602038675316462527073511653, −14.11559519604394364872780950453, −11.88489489985841879705847187495, −9.928291876343573439654977636493, −8.604610716410071406819271928209, −7.02760803773802445340981202045, −5.99290993035147340619415410330, −1.60609771259542928919537098829,
0.04327851414063456544186274605, 1.92506779331737391480357866069, 4.09090840347435161067658159326, 7.40363816377791698365652537736, 9.299080910708854285526467954159, 10.18485023088980871625455349509, 11.35341971667615104113110106854, 12.99546827134071927358123432223, 15.61498284361699966615753597861, 17.03660655975107305345210270084
