| L(s) = 1 | + (−2.04e3 − 3.54e3i)2-s + (9.12e5 − 1.22e5i)3-s + (−8.38e6 + 1.45e7i)4-s + (−3.58e8 + 6.20e8i)5-s + (−2.30e9 − 2.98e9i)6-s + (2.51e9 + 4.35e9i)7-s + (6.87e10 − 7.62e−6i)8-s + (8.17e11 − 2.23e11i)9-s + 2.93e12·10-s + (−8.96e12 − 1.55e13i)11-s + (−5.87e12 + 1.42e13i)12-s + (−4.35e13 + 7.55e13i)13-s + (1.02e13 − 1.78e13i)14-s + (−2.51e14 + 6.10e14i)15-s + (−1.40e14 − 2.43e14i)16-s − 5.86e14·17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.991 − 0.133i)3-s + (−0.249 + 0.433i)4-s + (−0.656 + 1.13i)5-s + (−0.431 − 0.559i)6-s + (0.0686 + 0.118i)7-s + 0.353·8-s + (0.964 − 0.263i)9-s + 0.928·10-s + (−0.861 − 1.49i)11-s + (−0.190 + 0.462i)12-s + (−0.518 + 0.898i)13-s + (0.0485 − 0.0840i)14-s + (−0.499 + 1.21i)15-s + (−0.125 − 0.216i)16-s − 0.244·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 + 0.904i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(1.777218081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.777218081\) |
| \(L(\frac{27}{2})\) |
|
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 | \( 1 + (-9.12e5 + 1.22e5i)T \) |
| good | 5 | \( 1 + (3.58e8 - 6.20e8i)T + (-1.49e17 - 2.58e17i)T^{2} \) |
| 7 | \( 1 + (-2.51e9 - 4.35e9i)T + (-6.70e20 + 1.16e21i)T^{2} \) |
| 11 | \( 1 + (8.96e12 + 1.55e13i)T + (-5.41e25 + 9.38e25i)T^{2} \) |
| 13 | \( 1 + (4.35e13 - 7.55e13i)T + (-3.52e27 - 6.11e27i)T^{2} \) |
| 17 | \( 1 + 5.86e14T + 5.77e30T^{2} \) |
| 19 | \( 1 + 8.89e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + (-7.76e16 + 1.34e17i)T + (-5.52e33 - 9.56e33i)T^{2} \) |
| 29 | \( 1 + (-4.42e17 - 7.65e17i)T + (-1.81e36 + 3.14e36i)T^{2} \) |
| 31 | \( 1 + (-9.67e17 + 1.67e18i)T + (-9.61e36 - 1.66e37i)T^{2} \) |
| 37 | \( 1 - 6.94e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + (1.80e19 - 3.12e19i)T + (-1.04e40 - 1.80e40i)T^{2} \) |
| 43 | \( 1 + (-2.04e19 - 3.53e19i)T + (-3.43e40 + 5.94e40i)T^{2} \) |
| 47 | \( 1 + (-3.90e20 - 6.75e20i)T + (-3.17e41 + 5.49e41i)T^{2} \) |
| 53 | \( 1 - 5.30e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + (-2.32e21 + 4.02e21i)T + (-9.33e43 - 1.61e44i)T^{2} \) |
| 61 | \( 1 + (1.46e22 + 2.54e22i)T + (-2.14e44 + 3.72e44i)T^{2} \) |
| 67 | \( 1 + (4.02e22 - 6.96e22i)T + (-2.24e45 - 3.88e45i)T^{2} \) |
| 71 | \( 1 - 1.89e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 1.19e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + (-7.54e22 - 1.30e23i)T + (-1.37e47 + 2.38e47i)T^{2} \) |
| 83 | \( 1 + (8.02e23 + 1.38e24i)T + (-4.74e47 + 8.21e47i)T^{2} \) |
| 89 | \( 1 - 1.82e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + (3.45e24 + 5.99e24i)T + (-2.33e49 + 4.04e49i)T^{2} \) |
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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
−12.91281080056407019919474499189, −11.34768021711178573883645878230, −10.43673572595470207251722222629, −8.880770180088915378512210133589, −7.906411979331363192978735602682, −6.66594223516772660441114258051, −4.24962592588675113641554560301, −3.00909478816104180136419697675, −2.36190438402363672472156423801, −0.53700994642494676966734859349,
0.891620493674944485801432974307, 2.36139037795704783298696353753, 4.21235532796390109700139961443, 5.08738159984831051711548885458, 7.34078482360453590831109862936, 8.041016807592867691391281209931, 9.176644512924594704134723836251, 10.29745152682135412079308990493, 12.47168574282839267052062898196, 13.32189599278466136206971053277
