| L(s) = 1 | + (1.31e3 + 2.27e3i)2-s + (2.39e6 − 4.14e6i)3-s + (2.64e8 − 4.58e8i)4-s + (1.11e10 + 1.92e10i)5-s + 1.25e10·6-s + (−1.79e12 + 6.33e9i)7-s + 2.80e12·8-s + (−1.14e13 − 1.98e13i)9-s + (−2.92e13 + 5.06e13i)10-s + (−4.71e14 + 8.17e14i)11-s + (−1.26e15 − 2.19e15i)12-s + 1.71e16·13-s + (−2.37e15 − 4.07e15i)14-s + 1.06e17·15-s + (−1.38e17 − 2.40e17i)16-s + (−3.49e17 + 6.05e17i)17-s + ⋯ |
| L(s) = 1 | + (0.0567 + 0.0982i)2-s + (0.288 − 0.499i)3-s + (0.493 − 0.854i)4-s + (0.815 + 1.41i)5-s + 0.0654·6-s + (−0.999 + 0.00352i)7-s + 0.225·8-s + (−0.166 − 0.288i)9-s + (−0.0925 + 0.160i)10-s + (−0.374 + 0.648i)11-s + (−0.284 − 0.493i)12-s + 1.20·13-s + (−0.0570 − 0.0980i)14-s + 0.941·15-s + (−0.480 − 0.832i)16-s + (−0.503 + 0.872i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(0.8202992211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8202992211\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.39e6 + 4.14e6i)T \) |
| 7 | \( 1 + (1.79e12 - 6.33e9i)T \) |
| good | 2 | \( 1 + (-1.31e3 - 2.27e3i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (-1.11e10 - 1.92e10i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (4.71e14 - 8.17e14i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 - 1.71e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (3.49e17 - 6.05e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (1.03e18 + 1.79e18i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-1.85e19 - 3.21e19i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 + 1.53e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (6.47e20 - 1.12e21i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (5.31e22 + 9.20e22i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 + 1.82e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 5.69e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-1.09e24 - 1.89e24i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (4.51e24 - 7.82e24i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (-2.28e25 + 3.96e25i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (1.43e25 + 2.48e25i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (-6.26e25 + 1.08e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 + 1.13e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (6.80e26 - 1.17e27i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (2.71e27 + 4.70e27i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 + 2.75e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (-1.47e28 - 2.55e28i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 - 1.60e28T + 4.13e57T^{2} \) |
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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
−12.96972605786532011867821638012, −11.06686250890622573369052217854, −10.34304997518708493701322648367, −9.214370243386884398610495692250, −7.19283554503500167890765032432, −6.49613373407488719876813592433, −5.72073467212056389885229863843, −3.48693078296838688319856211122, −2.37458020633893683397352047266, −1.56886323374858585882498967906,
0.13671692349773247848895916917, 1.60092873866256686689821629951, 2.90900294732117201949548442185, 3.94020738747316706793703915077, 5.33490488363515792601216192009, 6.56421495485570658485584117658, 8.379039070797547880216261550232, 8.999430276609101442770329255295, 10.31721929362643975323378144202, 11.78998463261997712482781332938
