| L(s) = 1 | + (51.7 − 37.6i)2-s + (−663. − 2.04e3i)3-s + (1.26e3 − 3.89e3i)4-s + (1.88e4 + 1.37e4i)5-s + (−1.11e5 − 8.07e4i)6-s + (−1.30e5 + 4.02e5i)7-s + (−8.10e4 − 2.49e5i)8-s + (−2.43e6 + 1.76e6i)9-s + 1.49e6·10-s + (−4.59e6 − 3.65e6i)11-s − 8.78e6·12-s + (−1.96e7 + 1.42e7i)13-s + (8.37e6 + 2.57e7i)14-s + (1.54e7 − 4.76e7i)15-s + (−1.35e7 − 9.86e6i)16-s + (1.57e8 + 1.14e8i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.525 − 1.61i)3-s + (0.154 − 0.475i)4-s + (0.540 + 0.392i)5-s + (−0.972 − 0.706i)6-s + (−0.420 + 1.29i)7-s + (−0.109 − 0.336i)8-s + (−1.52 + 1.10i)9-s + 0.472·10-s + (−0.782 − 0.622i)11-s − 0.849·12-s + (−1.12 + 0.819i)13-s + (0.297 + 0.915i)14-s + (0.350 − 1.07i)15-s + (−0.202 − 0.146i)16-s + (1.57 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.5221486842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5221486842\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + (4.59e6 + 3.65e6i)T \) |
| good | 3 | \( 1 + (663. + 2.04e3i)T + (-1.28e6 + 9.37e5i)T^{2} \) |
| 5 | \( 1 + (-1.88e4 - 1.37e4i)T + (3.77e8 + 1.16e9i)T^{2} \) |
| 7 | \( 1 + (1.30e5 - 4.02e5i)T + (-7.83e10 - 5.69e10i)T^{2} \) |
| 13 | \( 1 + (1.96e7 - 1.42e7i)T + (9.35e13 - 2.88e14i)T^{2} \) |
| 17 | \( 1 + (-1.57e8 - 1.14e8i)T + (3.06e15 + 9.41e15i)T^{2} \) |
| 19 | \( 1 + (-1.47e7 - 4.53e7i)T + (-3.40e16 + 2.47e16i)T^{2} \) |
| 23 | \( 1 + 1.05e9T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-6.36e7 + 1.95e8i)T + (-8.30e18 - 6.03e18i)T^{2} \) |
| 31 | \( 1 + (-1.98e9 + 1.44e9i)T + (7.54e18 - 2.32e19i)T^{2} \) |
| 37 | \( 1 + (-1.36e9 + 4.20e9i)T + (-1.97e20 - 1.43e20i)T^{2} \) |
| 41 | \( 1 + (-6.83e9 - 2.10e10i)T + (-7.48e20 + 5.43e20i)T^{2} \) |
| 43 | \( 1 + 7.77e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-7.47e9 - 2.30e10i)T + (-4.41e21 + 3.20e21i)T^{2} \) |
| 53 | \( 1 + (-3.04e10 + 2.21e10i)T + (8.04e21 - 2.47e22i)T^{2} \) |
| 59 | \( 1 + (6.89e10 - 2.12e11i)T + (-8.49e22 - 6.17e22i)T^{2} \) |
| 61 | \( 1 + (7.81e10 + 5.68e10i)T + (5.00e22 + 1.53e23i)T^{2} \) |
| 67 | \( 1 + 7.23e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (1.37e12 + 1.00e12i)T + (3.60e23 + 1.10e24i)T^{2} \) |
| 73 | \( 1 + (3.91e11 - 1.20e12i)T + (-1.35e24 - 9.82e23i)T^{2} \) |
| 79 | \( 1 + (2.09e11 - 1.51e11i)T + (1.44e24 - 4.43e24i)T^{2} \) |
| 83 | \( 1 + (-2.96e12 - 2.15e12i)T + (2.74e24 + 8.43e24i)T^{2} \) |
| 89 | \( 1 - 2.15e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (6.81e12 - 4.95e12i)T + (2.07e25 - 6.40e25i)T^{2} \) |
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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
−14.67083101293127985627171981317, −13.57236511832905241679909416429, −12.37026820392091155167473086740, −11.88388018825995444562251811977, −10.06620060799025960710188036552, −7.978931502548493541389312401994, −6.27735961015866520680211998327, −5.63696003852316203563643886630, −2.69638913941065886104631661656, −1.75003302448783211586125554277,
0.14530174843060948762435702439, 3.26396919290377852795266261485, 4.69052513618030976103538060314, 5.50502701312180102038648655992, 7.50617998794987084941305537663, 9.829881040707923802713211895390, 10.23741018385102121537392455377, 12.07311126555635461186683384179, 13.60035667204027962663102408889, 14.84294602043699332674032578726
