| L(s) = 1 | + (−2.04e3 − 3.54e3i)2-s + (−5.39e5 − 7.45e5i)3-s + (−8.38e6 + 1.45e7i)4-s + (1.32e8 − 2.29e8i)5-s + (−1.54e9 + 3.44e9i)6-s + (1.71e10 + 2.97e10i)7-s + (6.87e10 − 7.62e−6i)8-s + (−2.64e11 + 8.04e11i)9-s + (−1.08e12 + 6.10e−5i)10-s + (2.36e12 + 4.08e12i)11-s + (1.53e13 − 1.58e12i)12-s + (−7.65e13 + 1.32e14i)13-s + (7.04e13 − 1.22e14i)14-s + (−2.43e14 + 2.50e13i)15-s + (−1.40e14 − 2.43e14i)16-s + 3.04e15·17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.586 − 0.810i)3-s + (−0.249 + 0.433i)4-s + (0.243 − 0.421i)5-s + (−0.288 + 0.645i)6-s + (0.469 + 0.813i)7-s + 0.353·8-s + (−0.312 + 0.949i)9-s − 0.343·10-s + (0.226 + 0.392i)11-s + (0.497 − 0.0512i)12-s + (−0.910 + 1.57i)13-s + (0.332 − 0.575i)14-s + (−0.483 + 0.0498i)15-s + (−0.125 − 0.216i)16-s + 1.26·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(0.3915281590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3915281590\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.04e3 + 3.54e3i)T \) |
| 3 | \( 1 + (5.39e5 + 7.45e5i)T \) |
| good | 5 | \( 1 + (-1.32e8 + 2.29e8i)T + (-1.49e17 - 2.58e17i)T^{2} \) |
| 7 | \( 1 + (-1.71e10 - 2.97e10i)T + (-6.70e20 + 1.16e21i)T^{2} \) |
| 11 | \( 1 + (-2.36e12 - 4.08e12i)T + (-5.41e25 + 9.38e25i)T^{2} \) |
| 13 | \( 1 + (7.65e13 - 1.32e14i)T + (-3.52e27 - 6.11e27i)T^{2} \) |
| 17 | \( 1 - 3.04e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 1.34e16T + 9.30e31T^{2} \) |
| 23 | \( 1 + (2.18e16 - 3.77e16i)T + (-5.52e33 - 9.56e33i)T^{2} \) |
| 29 | \( 1 + (1.77e18 + 3.06e18i)T + (-1.81e36 + 3.14e36i)T^{2} \) |
| 31 | \( 1 + (-3.31e18 + 5.74e18i)T + (-9.61e36 - 1.66e37i)T^{2} \) |
| 37 | \( 1 + 4.38e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + (1.03e19 - 1.79e19i)T + (-1.04e40 - 1.80e40i)T^{2} \) |
| 43 | \( 1 + (4.98e19 + 8.63e19i)T + (-3.43e40 + 5.94e40i)T^{2} \) |
| 47 | \( 1 + (2.23e20 + 3.86e20i)T + (-3.17e41 + 5.49e41i)T^{2} \) |
| 53 | \( 1 - 4.84e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + (-2.56e21 + 4.44e21i)T + (-9.33e43 - 1.61e44i)T^{2} \) |
| 61 | \( 1 + (8.18e21 + 1.41e22i)T + (-2.14e44 + 3.72e44i)T^{2} \) |
| 67 | \( 1 + (4.45e22 - 7.71e22i)T + (-2.24e45 - 3.88e45i)T^{2} \) |
| 71 | \( 1 + 1.00e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 3.45e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + (-1.58e23 - 2.74e23i)T + (-1.37e47 + 2.38e47i)T^{2} \) |
| 83 | \( 1 + (8.67e23 + 1.50e24i)T + (-4.74e47 + 8.21e47i)T^{2} \) |
| 89 | \( 1 + 9.46e23T + 5.42e48T^{2} \) |
| 97 | \( 1 + (-7.88e23 - 1.36e24i)T + (-2.33e49 + 4.04e49i)T^{2} \) |
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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
−12.09168090163474419976346164857, −11.60126902130925187404651249124, −9.872166626009200383033775008614, −8.602491116238234781614564228902, −7.26871245667756762793519670448, −5.74248726066773141900320950534, −4.43622599944143965056670393623, −2.22042387216385803892488542761, −1.65029253731734129015307160128, −0.13119129385853728472410262029,
1.02073448981644120424204558063, 3.21338137310135035229563017477, 4.71945945636216844013948767065, 5.79662157447443610313244663251, 7.14546361319695368110333398320, 8.555596664041089935546781933842, 10.30697702382837720869863097024, 10.55429794093702369410083147902, 12.42216129918762560660403434369, 14.30946104348550756539094003337
