| L(s) = 1 | + (−51.7 − 37.6i)2-s + (412. − 1.27e3i)3-s + (1.26e3 + 3.89e3i)4-s + (−4.12e4 + 2.99e4i)5-s + (−6.91e4 + 5.02e4i)6-s + (2.53e4 + 7.81e4i)7-s + (8.10e4 − 2.49e5i)8-s + (−1.54e5 − 1.12e5i)9-s + 3.26e6·10-s + (5.35e6 − 2.42e6i)11-s + 5.47e6·12-s + (1.23e6 + 8.95e5i)13-s + (1.62e6 − 5.00e6i)14-s + (2.10e7 + 6.47e7i)15-s + (−1.35e7 + 9.86e6i)16-s + (−5.78e7 + 4.20e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.326 − 1.00i)3-s + (0.154 + 0.475i)4-s + (−1.18 + 0.857i)5-s + (−0.605 + 0.439i)6-s + (0.0815 + 0.251i)7-s + (0.109 − 0.336i)8-s + (−0.0968 − 0.0703i)9-s + 1.03·10-s + (0.910 − 0.412i)11-s + 0.529·12-s + (0.0708 + 0.0514i)13-s + (0.0576 − 0.177i)14-s + (0.477 + 1.46i)15-s + (−0.202 + 0.146i)16-s + (−0.581 + 0.422i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.268015159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.268015159\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + (-5.35e6 + 2.42e6i)T \) |
| good | 3 | \( 1 + (-412. + 1.27e3i)T + (-1.28e6 - 9.37e5i)T^{2} \) |
| 5 | \( 1 + (4.12e4 - 2.99e4i)T + (3.77e8 - 1.16e9i)T^{2} \) |
| 7 | \( 1 + (-2.53e4 - 7.81e4i)T + (-7.83e10 + 5.69e10i)T^{2} \) |
| 13 | \( 1 + (-1.23e6 - 8.95e5i)T + (9.35e13 + 2.88e14i)T^{2} \) |
| 17 | \( 1 + (5.78e7 - 4.20e7i)T + (3.06e15 - 9.41e15i)T^{2} \) |
| 19 | \( 1 + (-4.23e7 + 1.30e8i)T + (-3.40e16 - 2.47e16i)T^{2} \) |
| 23 | \( 1 - 6.72e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (6.42e8 + 1.97e9i)T + (-8.30e18 + 6.03e18i)T^{2} \) |
| 31 | \( 1 + (3.30e9 + 2.40e9i)T + (7.54e18 + 2.32e19i)T^{2} \) |
| 37 | \( 1 + (5.55e9 + 1.71e10i)T + (-1.97e20 + 1.43e20i)T^{2} \) |
| 41 | \( 1 + (4.12e9 - 1.26e10i)T + (-7.48e20 - 5.43e20i)T^{2} \) |
| 43 | \( 1 - 6.18e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-1.03e10 + 3.17e10i)T + (-4.41e21 - 3.20e21i)T^{2} \) |
| 53 | \( 1 + (1.26e11 + 9.19e10i)T + (8.04e21 + 2.47e22i)T^{2} \) |
| 59 | \( 1 + (-1.46e11 - 4.50e11i)T + (-8.49e22 + 6.17e22i)T^{2} \) |
| 61 | \( 1 + (-4.33e11 + 3.14e11i)T + (5.00e22 - 1.53e23i)T^{2} \) |
| 67 | \( 1 - 1.43e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + (1.35e11 - 9.87e10i)T + (3.60e23 - 1.10e24i)T^{2} \) |
| 73 | \( 1 + (-4.80e9 - 1.47e10i)T + (-1.35e24 + 9.82e23i)T^{2} \) |
| 79 | \( 1 + (1.59e12 + 1.15e12i)T + (1.44e24 + 4.43e24i)T^{2} \) |
| 83 | \( 1 + (-2.40e12 + 1.74e12i)T + (2.74e24 - 8.43e24i)T^{2} \) |
| 89 | \( 1 - 8.39e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (2.53e12 + 1.84e12i)T + (2.07e25 + 6.40e25i)T^{2} \) |
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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
−14.59274377267143805123305863308, −13.09537782800166906078114689167, −11.79831254673651681240742122868, −10.92814857975150501996770297847, −8.909778974423952663260155658016, −7.65169971184144353480178428936, −6.73553933169469404402574552961, −3.78501761089629516874141561564, −2.30565950996711756628656056873, −0.67561627765234689795895168079,
1.01350036916520629374408315412, 3.75997469339616217239267182037, 4.83372409539156143120311358282, 7.13721041829410264431215978825, 8.598059509984369359897160994012, 9.484996418085098859981833754433, 11.01566592139163311117311073346, 12.41733931365953150190211752636, 14.44024436956816782800475950908, 15.52221144772085231141388222914
