| L(s) = 1 | + (−29.2 − 89.9i)2-s + (−1.58e3 − 1.15e3i)3-s + (−602. + 437. i)4-s + (−1.91e4 + 5.88e4i)5-s + (−5.71e4 + 1.76e5i)6-s + (1.19e5 − 8.68e4i)7-s + (−5.69e5 − 4.13e5i)8-s + (6.91e5 + 2.12e6i)9-s + 5.85e6·10-s + (−2.88e6 − 5.11e6i)11-s + 1.45e6·12-s + (3.06e6 + 9.42e6i)13-s + (−1.12e7 − 8.20e6i)14-s + (9.80e7 − 7.12e7i)15-s + (−2.24e7 + 6.90e7i)16-s + (1.20e7 − 3.71e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.322 − 0.993i)2-s + (−1.25 − 0.911i)3-s + (−0.0735 + 0.0534i)4-s + (−0.547 + 1.68i)5-s + (−0.500 + 1.54i)6-s + (0.383 − 0.278i)7-s + (−0.768 − 0.558i)8-s + (0.433 + 1.33i)9-s + 1.85·10-s + (−0.491 − 0.871i)11-s + 0.140·12-s + (0.176 + 0.541i)13-s + (−0.400 − 0.291i)14-s + (2.22 − 1.61i)15-s + (−0.334 + 1.02i)16-s + (0.121 − 0.373i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.621306 - 0.0500169i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.621306 - 0.0500169i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (2.88e6 + 5.11e6i)T \) |
| good | 2 | \( 1 + (29.2 + 89.9i)T + (-6.62e3 + 4.81e3i)T^{2} \) |
| 3 | \( 1 + (1.58e3 + 1.15e3i)T + (4.92e5 + 1.51e6i)T^{2} \) |
| 5 | \( 1 + (1.91e4 - 5.88e4i)T + (-9.87e8 - 7.17e8i)T^{2} \) |
| 7 | \( 1 + (-1.19e5 + 8.68e4i)T + (2.99e10 - 9.21e10i)T^{2} \) |
| 13 | \( 1 + (-3.06e6 - 9.42e6i)T + (-2.45e14 + 1.78e14i)T^{2} \) |
| 17 | \( 1 + (-1.20e7 + 3.71e7i)T + (-8.01e15 - 5.82e15i)T^{2} \) |
| 19 | \( 1 + (-2.54e8 - 1.84e8i)T + (1.29e16 + 3.99e16i)T^{2} \) |
| 23 | \( 1 - 2.30e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-1.07e9 + 7.84e8i)T + (3.17e18 - 9.75e18i)T^{2} \) |
| 31 | \( 1 + (-2.76e9 - 8.51e9i)T + (-1.97e19 + 1.43e19i)T^{2} \) |
| 37 | \( 1 + (5.21e9 - 3.78e9i)T + (7.52e19 - 2.31e20i)T^{2} \) |
| 41 | \( 1 + (-2.13e10 - 1.55e10i)T + (2.85e20 + 8.79e20i)T^{2} \) |
| 43 | \( 1 - 6.09e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (4.31e10 + 3.13e10i)T + (1.68e21 + 5.19e21i)T^{2} \) |
| 53 | \( 1 + (-5.58e8 - 1.71e9i)T + (-2.10e22 + 1.53e22i)T^{2} \) |
| 59 | \( 1 + (4.88e11 - 3.54e11i)T + (3.24e22 - 9.98e22i)T^{2} \) |
| 61 | \( 1 + (9.78e10 - 3.01e11i)T + (-1.30e23 - 9.51e22i)T^{2} \) |
| 67 | \( 1 - 2.23e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-5.88e10 + 1.81e11i)T + (-9.42e23 - 6.84e23i)T^{2} \) |
| 73 | \( 1 + (6.96e11 - 5.06e11i)T + (5.16e23 - 1.59e24i)T^{2} \) |
| 79 | \( 1 + (5.59e11 + 1.72e12i)T + (-3.77e24 + 2.74e24i)T^{2} \) |
| 83 | \( 1 + (6.36e11 - 1.95e12i)T + (-7.17e24 - 5.21e24i)T^{2} \) |
| 89 | \( 1 + 9.81e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + (6.90e11 + 2.12e12i)T + (-5.44e25 + 3.95e25i)T^{2} \) |
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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
−17.94916058857230893818599309400, −15.97529161374361232872455543008, −14.07125382108149956761556758586, −12.04189086317161769756692590953, −11.24607308296827908346918288473, −10.47853309175443271924575094955, −7.36440172640674748424368426011, −6.14202700110369741004483196153, −3.03057243722127971002442348450, −1.12966245492675166719047928470,
0.43124502129406916751169101563, 4.69935636735707968160457091800, 5.59478171535600415689477302606, 7.83200814222675017122936307070, 9.359464269920169481771868040037, 11.42555733991569089098403824982, 12.52363100989065795903057197775, 15.45888512866149862170622007328, 15.90730054253346099059023171504, 17.04053823834881144327038946652
