L(s) = 1 | + (−3.95e4 + 3.95e4i)2-s + (4.69e7 + 4.69e7i)3-s + 1.16e9i·4-s + (−7.61e10 − 1.32e11i)5-s − 3.71e12·6-s + (2.85e13 − 2.85e13i)7-s + (−2.16e14 − 2.16e14i)8-s + 2.55e15i·9-s + (8.23e15 + 2.21e15i)10-s + 6.29e16·11-s + (−5.48e16 + 5.48e16i)12-s + (6.68e17 + 6.68e17i)13-s + 2.25e18i·14-s + (2.63e18 − 9.78e18i)15-s + 1.20e19·16-s + (9.86e17 − 9.86e17i)17-s + ⋯ |
L(s) = 1 | + (−0.603 + 0.603i)2-s + (1.09 + 1.09i)3-s + 0.272i·4-s + (−0.499 − 0.866i)5-s − 1.31·6-s + (0.858 − 0.858i)7-s + (−0.767 − 0.767i)8-s + 1.37i·9-s + (0.823 + 0.221i)10-s + 1.36·11-s + (−0.296 + 0.296i)12-s + (1.00 + 1.00i)13-s + 1.03i·14-s + (0.400 − 1.48i)15-s + 0.653·16-s + (0.0202 − 0.0202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0307 - 0.999i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.0307 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(2.374294459\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.374294459\) |
\(L(17)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (7.61e10 + 1.32e11i)T \) |
good | 2 | \( 1 + (3.95e4 - 3.95e4i)T - 4.29e9iT^{2} \) |
| 3 | \( 1 + (-4.69e7 - 4.69e7i)T + 1.85e15iT^{2} \) |
| 7 | \( 1 + (-2.85e13 + 2.85e13i)T - 1.10e27iT^{2} \) |
| 11 | \( 1 - 6.29e16T + 2.11e33T^{2} \) |
| 13 | \( 1 + (-6.68e17 - 6.68e17i)T + 4.42e35iT^{2} \) |
| 17 | \( 1 + (-9.86e17 + 9.86e17i)T - 2.36e39iT^{2} \) |
| 19 | \( 1 + 3.77e20iT - 8.31e40T^{2} \) |
| 23 | \( 1 + (-5.92e21 - 5.92e21i)T + 3.76e43iT^{2} \) |
| 29 | \( 1 - 2.77e19iT - 6.26e46T^{2} \) |
| 31 | \( 1 + 3.93e23T + 5.29e47T^{2} \) |
| 37 | \( 1 + (4.19e24 - 4.19e24i)T - 1.52e50iT^{2} \) |
| 41 | \( 1 - 8.53e25T + 4.06e51T^{2} \) |
| 43 | \( 1 + (8.89e25 + 8.89e25i)T + 1.86e52iT^{2} \) |
| 47 | \( 1 + (-3.07e26 + 3.07e26i)T - 3.21e53iT^{2} \) |
| 53 | \( 1 + (-4.54e26 - 4.54e26i)T + 1.50e55iT^{2} \) |
| 59 | \( 1 - 1.68e28iT - 4.64e56T^{2} \) |
| 61 | \( 1 + 1.17e28T + 1.35e57T^{2} \) |
| 67 | \( 1 + (1.91e29 - 1.91e29i)T - 2.71e58iT^{2} \) |
| 71 | \( 1 + 1.41e29T + 1.73e59T^{2} \) |
| 73 | \( 1 + (-5.53e29 - 5.53e29i)T + 4.22e59iT^{2} \) |
| 79 | \( 1 + 1.90e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 + (-2.78e30 - 2.78e30i)T + 2.57e61iT^{2} \) |
| 89 | \( 1 - 2.46e30iT - 2.40e62T^{2} \) |
| 97 | \( 1 + (-4.72e31 + 4.72e31i)T - 3.77e63iT^{2} \) |
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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
−16.31876585012075478731780234893, −15.14618346542925912493419386213, −13.67974019418091258020397434910, −11.41583079108673792590652169717, −9.178905802867691918955164154774, −8.722024969221625184090077291070, −7.25592639326110641666780781164, −4.39368386534366931055303787208, −3.64666913858317806463352838856, −1.13493373264332515013116975939,
1.04715098117995375637562745237, 2.03156842782430964661062216484, 3.26766756781750666141263797390, 6.20385437567295947610059120678, 7.973045833102725641035011370665, 8.930549038402896780723723736713, 10.93235250356633781986512347391, 12.23637273658500048664917437814, 14.33462198891936668488200246170, 14.91956353007382474742821786077