| L(s) = 1 | + (−2.04e3 + 3.54e3i)2-s + (4.51e5 − 8.01e5i)3-s + (−8.38e6 − 1.45e7i)4-s + (−2.81e7 − 4.87e7i)5-s + (1.91e9 + 3.24e9i)6-s + (2.24e10 − 3.89e10i)7-s + (6.87e10 + 7.62e−6i)8-s + (−4.38e11 − 7.24e11i)9-s + 2.30e11·10-s + (2.18e12 − 3.78e12i)11-s + (−1.54e13 + 1.60e11i)12-s + (−5.98e13 − 1.03e14i)13-s + (9.21e13 + 1.59e14i)14-s + (−5.18e13 + 5.39e11i)15-s + (−1.40e14 + 2.43e14i)16-s − 2.36e15·17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.490 − 0.871i)3-s + (−0.249 − 0.433i)4-s + (−0.0515 − 0.0893i)5-s + (0.359 + 0.608i)6-s + (0.614 − 1.06i)7-s + 0.353·8-s + (−0.517 − 0.855i)9-s + 0.0729·10-s + (0.209 − 0.363i)11-s + (−0.499 + 0.00519i)12-s + (−0.712 − 1.23i)13-s + (0.434 + 0.752i)14-s + (−0.103 + 0.00107i)15-s + (−0.125 + 0.216i)16-s − 0.983·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(1.538696628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.538696628\) |
| \(L(\frac{27}{2})\) |
|
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 | \( 1 + (-4.51e5 + 8.01e5i)T \) |
| good | 5 | \( 1 + (2.81e7 + 4.87e7i)T + (-1.49e17 + 2.58e17i)T^{2} \) |
| 7 | \( 1 + (-2.24e10 + 3.89e10i)T + (-6.70e20 - 1.16e21i)T^{2} \) |
| 11 | \( 1 + (-2.18e12 + 3.78e12i)T + (-5.41e25 - 9.38e25i)T^{2} \) |
| 13 | \( 1 + (5.98e13 + 1.03e14i)T + (-3.52e27 + 6.11e27i)T^{2} \) |
| 17 | \( 1 + 2.36e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 1.56e16T + 9.30e31T^{2} \) |
| 23 | \( 1 + (4.68e16 + 8.11e16i)T + (-5.52e33 + 9.56e33i)T^{2} \) |
| 29 | \( 1 + (1.23e17 - 2.13e17i)T + (-1.81e36 - 3.14e36i)T^{2} \) |
| 31 | \( 1 + (3.54e17 + 6.13e17i)T + (-9.61e36 + 1.66e37i)T^{2} \) |
| 37 | \( 1 - 6.90e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + (-1.39e20 - 2.41e20i)T + (-1.04e40 + 1.80e40i)T^{2} \) |
| 43 | \( 1 + (1.91e20 - 3.30e20i)T + (-3.43e40 - 5.94e40i)T^{2} \) |
| 47 | \( 1 + (-2.48e20 + 4.31e20i)T + (-3.17e41 - 5.49e41i)T^{2} \) |
| 53 | \( 1 + 4.77e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + (6.60e21 + 1.14e22i)T + (-9.33e43 + 1.61e44i)T^{2} \) |
| 61 | \( 1 + (-1.61e21 + 2.80e21i)T + (-2.14e44 - 3.72e44i)T^{2} \) |
| 67 | \( 1 + (6.67e21 + 1.15e22i)T + (-2.24e45 + 3.88e45i)T^{2} \) |
| 71 | \( 1 + 1.70e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 1.13e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + (2.90e22 - 5.03e22i)T + (-1.37e47 - 2.38e47i)T^{2} \) |
| 83 | \( 1 + (-2.90e23 + 5.04e23i)T + (-4.74e47 - 8.21e47i)T^{2} \) |
| 89 | \( 1 + 2.58e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + (-3.12e24 + 5.41e24i)T + (-2.33e49 - 4.04e49i)T^{2} \) |
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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
−12.86312156438471413118817106978, −11.22978804463419529905289127313, −9.707372312049927483516745893761, −8.166999598066624054847005458194, −7.52896336114656467844791078847, −6.25599182459464727663226721521, −4.61139049848012890796107409554, −2.86117679428463153870102676431, −1.19274495391294558698246739755, −0.41081811297475326036976929697,
1.74854346693616459654178236965, 2.69349239297067578461101215124, 4.14761334824981298291621976263, 5.28796684473049411708108039192, 7.49935361823417158275702399052, 8.988916276402611747181550633585, 9.537329008435950078468117003938, 11.15556658630959378590569910218, 11.98530466357684827635648973054, 13.80181820593942215322559926303
