| L(s) = 1 | + (−204. + 117. i)2-s + (8.39e3 − 7.65e3i)3-s + (−3.77e4 + 6.53e4i)4-s + (−1.27e5 − 2.20e5i)5-s + (−8.12e5 + 2.55e6i)6-s + (−1.34e7 + 7.12e6i)7-s − 4.87e7i·8-s + (1.19e7 − 1.28e8i)9-s + (5.19e7 + 2.99e7i)10-s + (−3.53e7 − 2.04e7i)11-s + (1.83e8 + 8.37e8i)12-s + 3.27e9i·13-s + (1.91e9 − 3.04e9i)14-s + (−2.75e9 − 8.77e8i)15-s + (7.95e8 + 1.37e9i)16-s + (4.16e9 − 7.21e9i)17-s + ⋯ |
| L(s) = 1 | + (−0.564 + 0.325i)2-s + (0.739 − 0.673i)3-s + (−0.287 + 0.498i)4-s + (−0.145 − 0.252i)5-s + (−0.197 + 0.620i)6-s + (−0.884 + 0.467i)7-s − 1.02i·8-s + (0.0925 − 0.995i)9-s + (0.164 + 0.0948i)10-s + (−0.0497 − 0.0287i)11-s + (0.123 + 0.562i)12-s + 1.11i·13-s + (0.346 − 0.551i)14-s + (−0.277 − 0.0883i)15-s + (0.0462 + 0.0801i)16-s + (0.144 − 0.250i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.360117198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.360117198\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-8.39e3 + 7.65e3i)T \) |
| 7 | \( 1 + (1.34e7 - 7.12e6i)T \) |
| good | 2 | \( 1 + (204. - 117. i)T + (6.55e4 - 1.13e5i)T^{2} \) |
| 5 | \( 1 + (1.27e5 + 2.20e5i)T + (-3.81e11 + 6.60e11i)T^{2} \) |
| 11 | \( 1 + (3.53e7 + 2.04e7i)T + (2.52e17 + 4.37e17i)T^{2} \) |
| 13 | \( 1 - 3.27e9iT - 8.65e18T^{2} \) |
| 17 | \( 1 + (-4.16e9 + 7.21e9i)T + (-4.13e20 - 7.16e20i)T^{2} \) |
| 19 | \( 1 + (-1.80e10 + 1.04e10i)T + (2.74e21 - 4.74e21i)T^{2} \) |
| 23 | \( 1 + (1.71e11 - 9.92e10i)T + (7.05e22 - 1.22e23i)T^{2} \) |
| 29 | \( 1 - 8.76e11iT - 7.25e24T^{2} \) |
| 31 | \( 1 + (1.91e12 + 1.10e12i)T + (1.12e25 + 1.95e25i)T^{2} \) |
| 37 | \( 1 + (-1.31e13 - 2.26e13i)T + (-2.28e26 + 3.95e26i)T^{2} \) |
| 41 | \( 1 - 8.11e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 7.57e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + (-5.57e13 - 9.65e13i)T + (-1.33e28 + 2.30e28i)T^{2} \) |
| 53 | \( 1 + (-7.02e14 - 4.05e14i)T + (1.02e29 + 1.77e29i)T^{2} \) |
| 59 | \( 1 + (-4.44e14 + 7.69e14i)T + (-6.35e29 - 1.10e30i)T^{2} \) |
| 61 | \( 1 + (1.50e14 - 8.68e13i)T + (1.12e30 - 1.94e30i)T^{2} \) |
| 67 | \( 1 + (-7.69e14 + 1.33e15i)T + (-5.52e30 - 9.56e30i)T^{2} \) |
| 71 | \( 1 + 1.97e15iT - 2.96e31T^{2} \) |
| 73 | \( 1 + (-6.07e15 - 3.51e15i)T + (2.37e31 + 4.11e31i)T^{2} \) |
| 79 | \( 1 + (-4.11e15 - 7.13e15i)T + (-9.09e31 + 1.57e32i)T^{2} \) |
| 83 | \( 1 - 1.52e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + (2.16e16 + 3.75e16i)T + (-6.89e32 + 1.19e33i)T^{2} \) |
| 97 | \( 1 - 5.26e16iT - 5.95e33T^{2} \) |
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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
−14.07737843275622235735314540090, −12.91932721727485131558072730754, −12.02864833359123039050662245017, −9.564175996207389763888191672772, −8.833444416519144251818852202580, −7.57642376744981678560315850548, −6.42614648596826513437478776568, −3.99385060835455075232975682278, −2.61419667347427242792754775190, −0.76944189736347365255285215974,
0.68382263641435516344731378937, 2.52283443334658237830913745460, 3.87391141878292419274083055374, 5.57950618410590922827310252612, 7.66162046555514385124124614367, 9.050842144334668681260778480570, 10.07617316046487042389655535972, 10.84409906616903602434594473860, 13.00097468354834972881547478597, 14.24518325291366840232119470978
