| L(s) = 1 | + (−1.75e4 − 3.04e4i)2-s + (−2.39e6 + 4.14e6i)3-s + (−3.48e8 + 6.03e8i)4-s + (−6.19e9 − 1.07e10i)5-s + 1.67e11·6-s + (8.11e11 + 1.60e12i)7-s + (5.61e12 − 0.00195i)8-s + (−1.14e13 − 1.98e13i)9-s + (−2.17e14 + 3.76e14i)10-s + (−1.12e14 + 1.94e14i)11-s + (−1.66e15 − 2.88e15i)12-s + 1.23e16·13-s + (3.44e16 − 5.27e16i)14-s + 5.92e16·15-s + (8.83e16 + 1.53e17i)16-s + (−5.23e17 + 9.06e17i)17-s + ⋯ |
| L(s) = 1 | + (−0.757 − 1.31i)2-s + (−0.288 + 0.499i)3-s + (−0.648 + 1.12i)4-s + (−0.453 − 0.785i)5-s + 0.875·6-s + (0.452 + 0.891i)7-s + 0.451·8-s + (−0.166 − 0.288i)9-s + (−0.687 + 1.19i)10-s + (−0.0891 + 0.154i)11-s + (−0.374 − 0.648i)12-s + 0.873·13-s + (0.827 − 1.26i)14-s + 0.523·15-s + (0.306 + 0.531i)16-s + (−0.753 + 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(0.4012478614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4012478614\) |
| \(L(\frac{31}{2})\) |
|
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 | \( 1 + (-8.11e11 - 1.60e12i)T \) |
| good | 2 | \( 1 + (1.75e4 + 3.04e4i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (6.19e9 + 1.07e10i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (1.12e14 - 1.94e14i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 - 1.23e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (5.23e17 - 9.06e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (-6.50e17 - 1.12e18i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-1.01e19 - 1.75e19i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 - 1.23e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (1.65e20 - 2.86e20i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (-4.09e21 - 7.09e21i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 - 7.25e22T + 5.89e46T^{2} \) |
| 43 | \( 1 + 1.02e22T + 2.34e47T^{2} \) |
| 47 | \( 1 + (1.44e24 + 2.50e24i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (-3.52e24 + 6.10e24i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (1.81e23 - 3.13e23i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (3.51e25 + 6.09e25i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (1.04e26 - 1.80e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 + 1.02e27T + 4.85e53T^{2} \) |
| 73 | \( 1 + (7.59e26 - 1.31e27i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (4.98e26 + 8.62e26i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 - 7.24e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (5.50e27 + 9.54e27i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 - 7.69e28T + 4.13e57T^{2} \) |
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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
−12.02645923348880788684721942109, −11.21014672805356517730864970350, −10.14930790538958294259953481416, −8.803028668383113771026546443914, −8.374993209714056003252594774474, −5.98785390281610517786775929563, −4.57571796620757461721634069080, −3.45247440522659583935056013712, −2.03055153148811029846477244418, −1.06525855014827113976299893598,
0.14965305303771457694917046231, 1.07694353906361635356695990646, 2.99174777443270317071816974759, 4.71632219306936426298564527461, 6.22625650505224874399107898785, 7.09294325259961955514605879925, 7.75986947619555393237504540583, 8.995439289684606311754947757851, 10.61216984739987827921652659148, 11.56778966631601505848931107765
