| L(s) = 1 | + (−9.51e4 − 5.49e4i)2-s + (4.10e7 − 1.30e7i)3-s + (3.88e9 + 6.73e9i)4-s + (−1.11e11 − 6.45e10i)5-s + (−4.62e12 − 1.00e12i)6-s + (2.02e13 − 2.63e13i)7-s + (0.0625 − 3.82e14i)8-s + (1.51e15 − 1.07e15i)9-s + (7.09e15 + 1.22e16i)10-s + (1.13e16 − 6.54e15i)11-s + (2.47e17 + 2.25e17i)12-s − 3.93e17·13-s + (−3.37e18 + 1.38e18i)14-s + (−5.43e18 − 1.18e18i)15-s + (−4.30e18 + 7.45e18i)16-s + (2.41e18 − 1.39e18i)17-s + ⋯ |
| L(s) = 1 | + (−1.45 − 0.838i)2-s + (0.952 − 0.303i)3-s + (0.905 + 1.56i)4-s + (−0.733 − 0.423i)5-s + (−1.63 − 0.357i)6-s + (0.610 − 0.791i)7-s − 1.35i·8-s + (0.815 − 0.578i)9-s + (0.709 + 1.22i)10-s + (0.246 − 0.142i)11-s + (1.33 + 1.21i)12-s − 0.591·13-s + (−1.55 + 0.637i)14-s + (−0.826 − 0.180i)15-s + (−0.233 + 0.404i)16-s + (0.0495 − 0.0286i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.280i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(1.322808290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.322808290\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.10e7 + 1.30e7i)T \) |
| 7 | \( 1 + (-2.02e13 + 2.63e13i)T \) |
| good | 2 | \( 1 + (9.51e4 + 5.49e4i)T + (2.14e9 + 3.71e9i)T^{2} \) |
| 5 | \( 1 + (1.11e11 + 6.45e10i)T + (1.16e22 + 2.01e22i)T^{2} \) |
| 11 | \( 1 + (-1.13e16 + 6.54e15i)T + (1.05e33 - 1.82e33i)T^{2} \) |
| 13 | \( 1 + 3.93e17T + 4.42e35T^{2} \) |
| 17 | \( 1 + (-2.41e18 + 1.39e18i)T + (1.18e39 - 2.05e39i)T^{2} \) |
| 19 | \( 1 + (-2.34e20 + 4.06e20i)T + (-4.15e40 - 7.20e40i)T^{2} \) |
| 23 | \( 1 + (-3.57e21 - 2.06e21i)T + (1.88e43 + 3.25e43i)T^{2} \) |
| 29 | \( 1 - 9.20e22iT - 6.26e46T^{2} \) |
| 31 | \( 1 + (4.13e23 + 7.16e23i)T + (-2.64e47 + 4.58e47i)T^{2} \) |
| 37 | \( 1 + (-4.88e24 + 8.45e24i)T + (-7.61e49 - 1.31e50i)T^{2} \) |
| 41 | \( 1 - 9.10e25iT - 4.06e51T^{2} \) |
| 43 | \( 1 - 5.26e25T + 1.86e52T^{2} \) |
| 47 | \( 1 + (-6.86e26 - 3.96e26i)T + (1.60e53 + 2.78e53i)T^{2} \) |
| 53 | \( 1 + (-1.42e27 + 8.21e26i)T + (7.51e54 - 1.30e55i)T^{2} \) |
| 59 | \( 1 + (-1.27e28 + 7.33e27i)T + (2.32e56 - 4.02e56i)T^{2} \) |
| 61 | \( 1 + (1.94e27 - 3.37e27i)T + (-6.75e56 - 1.16e57i)T^{2} \) |
| 67 | \( 1 + (1.10e28 + 1.92e28i)T + (-1.35e58 + 2.35e58i)T^{2} \) |
| 71 | \( 1 + 1.54e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + (-1.70e29 - 2.95e29i)T + (-2.11e59 + 3.66e59i)T^{2} \) |
| 79 | \( 1 + (-2.00e30 + 3.48e30i)T + (-2.64e60 - 4.58e60i)T^{2} \) |
| 83 | \( 1 + 9.63e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + (5.98e30 + 3.45e30i)T + (1.20e62 + 2.07e62i)T^{2} \) |
| 97 | \( 1 - 7.70e31T + 3.77e63T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.01616773225091490597264809270, −9.658572442634525198464075259538, −8.842913219061199922832898421032, −7.75516893109344979922352316950, −7.26558569258691135229090613961, −4.52331937175749229964646070530, −3.30410507345799561989099071058, −2.20621192152202584649592277016, −1.03715144268065268939523602040, −0.47431031199559176915132878194,
1.18379028479871705590712478712, 2.30598886353181208807146487710, 3.74568446809028587177222358029, 5.40654998097500521103001222432, 7.07970008132666206229730057767, 7.82029228231258011450595186477, 8.667277961843172719857008728863, 9.612937915918492246766832252537, 10.70924826203373571370411940718, 12.16500865072147402123591103268
