| L(s) = 1 | + (51.7 − 37.6i)2-s + (460. + 1.41e3i)3-s + (1.26e3 − 3.89e3i)4-s + (4.68e4 + 3.40e4i)5-s + (7.71e4 + 5.60e4i)6-s + (−1.57e5 + 4.84e5i)7-s + (−8.10e4 − 2.49e5i)8-s + (−5.04e5 + 3.66e5i)9-s + 3.70e6·10-s + (−5.82e6 − 7.67e5i)11-s + 6.09e6·12-s + (1.21e7 − 8.85e6i)13-s + (1.00e7 + 3.10e7i)14-s + (−2.66e7 + 8.19e7i)15-s + (−1.35e7 − 9.86e6i)16-s + (−1.40e8 − 1.01e8i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.364 + 1.12i)3-s + (0.154 − 0.475i)4-s + (1.34 + 0.973i)5-s + (0.674 + 0.490i)6-s + (−0.505 + 1.55i)7-s + (−0.109 − 0.336i)8-s + (−0.316 + 0.229i)9-s + 1.17·10-s + (−0.991 − 0.130i)11-s + 0.589·12-s + (0.700 − 0.508i)13-s + (0.357 + 1.10i)14-s + (−0.603 + 1.85i)15-s + (−0.202 − 0.146i)16-s + (−1.41 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(3.320451457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.320451457\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + (5.82e6 + 7.67e5i)T \) |
| good | 3 | \( 1 + (-460. - 1.41e3i)T + (-1.28e6 + 9.37e5i)T^{2} \) |
| 5 | \( 1 + (-4.68e4 - 3.40e4i)T + (3.77e8 + 1.16e9i)T^{2} \) |
| 7 | \( 1 + (1.57e5 - 4.84e5i)T + (-7.83e10 - 5.69e10i)T^{2} \) |
| 13 | \( 1 + (-1.21e7 + 8.85e6i)T + (9.35e13 - 2.88e14i)T^{2} \) |
| 17 | \( 1 + (1.40e8 + 1.01e8i)T + (3.06e15 + 9.41e15i)T^{2} \) |
| 19 | \( 1 + (-5.80e7 - 1.78e8i)T + (-3.40e16 + 2.47e16i)T^{2} \) |
| 23 | \( 1 - 2.30e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (7.50e8 - 2.30e9i)T + (-8.30e18 - 6.03e18i)T^{2} \) |
| 31 | \( 1 + (-3.92e9 + 2.84e9i)T + (7.54e18 - 2.32e19i)T^{2} \) |
| 37 | \( 1 + (3.31e9 - 1.02e10i)T + (-1.97e20 - 1.43e20i)T^{2} \) |
| 41 | \( 1 + (-9.68e8 - 2.98e9i)T + (-7.48e20 + 5.43e20i)T^{2} \) |
| 43 | \( 1 - 6.46e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-1.54e10 - 4.74e10i)T + (-4.41e21 + 3.20e21i)T^{2} \) |
| 53 | \( 1 + (-1.01e11 + 7.40e10i)T + (8.04e21 - 2.47e22i)T^{2} \) |
| 59 | \( 1 + (-8.18e10 + 2.51e11i)T + (-8.49e22 - 6.17e22i)T^{2} \) |
| 61 | \( 1 + (4.99e11 + 3.62e11i)T + (5.00e22 + 1.53e23i)T^{2} \) |
| 67 | \( 1 - 8.79e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (3.79e11 + 2.75e11i)T + (3.60e23 + 1.10e24i)T^{2} \) |
| 73 | \( 1 + (-4.45e9 + 1.37e10i)T + (-1.35e24 - 9.82e23i)T^{2} \) |
| 79 | \( 1 + (1.36e12 - 9.91e11i)T + (1.44e24 - 4.43e24i)T^{2} \) |
| 83 | \( 1 + (-3.21e12 - 2.33e12i)T + (2.74e24 + 8.43e24i)T^{2} \) |
| 89 | \( 1 + 1.94e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + (-8.77e12 + 6.37e12i)T + (2.07e25 - 6.40e25i)T^{2} \) |
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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
−15.28236683446565325701297092506, −14.03756132529471375651089343099, −12.88256726039127251465570462679, −11.02445764840076250614823866567, −9.980268253974658985178541407536, −9.067222858795985828092356499770, −6.25267384263560186377168633649, −5.20726403719484832824693286599, −3.11319567167157830002217609484, −2.37008374166453009352948456811,
0.888848818707467656064778949268, 2.22184207229717048610825155155, 4.44430805535753297870735296322, 6.18621756682462171495943691931, 7.27646916492502247932025648744, 8.792495958136898331638277980340, 10.52515372622542546266397244552, 12.81600000011495539786688044150, 13.42123589678487427654440556758, 13.75514493305049419246663101060
