| L(s) = 1 | − 7.13e4i·2-s + 3.08e7i·3-s − 2.93e9·4-s + 2.20e12·6-s − 1.14e13i·7-s + 5.63e13i·8-s − 3.34e14·9-s − 2.40e15·11-s − 9.06e16i·12-s − 2.01e17i·13-s − 8.15e17·14-s − 2.29e18·16-s − 1.09e19i·17-s + 2.38e19i·18-s + 1.42e19·19-s + ⋯ |
| L(s) = 1 | − 1.53i·2-s + 1.24i·3-s − 1.36·4-s + 1.91·6-s − 0.910i·7-s + 0.565i·8-s − 0.541·9-s − 0.173·11-s − 1.69i·12-s − 1.09i·13-s − 1.40·14-s − 0.496·16-s − 0.925i·17-s + 0.832i·18-s + 0.215·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(16)\) |
\(\approx\) |
\(0.7896246655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7896246655\) |
| \(L(\frac{33}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 7.13e4iT - 2.14e9T^{2} \) |
| 3 | \( 1 - 3.08e7iT - 6.17e14T^{2} \) |
| 7 | \( 1 + 1.14e13iT - 1.57e26T^{2} \) |
| 11 | \( 1 + 2.40e15T + 1.91e32T^{2} \) |
| 13 | \( 1 + 2.01e17iT - 3.40e34T^{2} \) |
| 17 | \( 1 + 1.09e19iT - 1.39e38T^{2} \) |
| 19 | \( 1 - 1.42e19T + 4.37e39T^{2} \) |
| 23 | \( 1 + 3.85e18iT - 1.63e42T^{2} \) |
| 29 | \( 1 + 7.63e22T + 2.15e45T^{2} \) |
| 31 | \( 1 - 1.86e23T + 1.70e46T^{2} \) |
| 37 | \( 1 + 1.23e24iT - 4.11e48T^{2} \) |
| 41 | \( 1 - 1.38e25T + 9.91e49T^{2} \) |
| 43 | \( 1 + 2.67e25iT - 4.34e50T^{2} \) |
| 47 | \( 1 + 7.40e25iT - 6.83e51T^{2} \) |
| 53 | \( 1 - 3.56e25iT - 2.83e53T^{2} \) |
| 59 | \( 1 - 2.36e27T + 7.87e54T^{2} \) |
| 61 | \( 1 + 5.44e27T + 2.21e55T^{2} \) |
| 67 | \( 1 - 9.41e27iT - 4.05e56T^{2} \) |
| 71 | \( 1 + 2.10e28T + 2.44e57T^{2} \) |
| 73 | \( 1 - 3.92e28iT - 5.79e57T^{2} \) |
| 79 | \( 1 + 1.79e29T + 6.70e58T^{2} \) |
| 83 | \( 1 + 4.54e29iT - 3.10e59T^{2} \) |
| 89 | \( 1 + 2.60e29T + 2.69e60T^{2} \) |
| 97 | \( 1 - 5.38e30iT - 3.88e61T^{2} \) |
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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
−10.57487794672674399583095010692, −10.03871514969534162504272551127, −9.070973617009258277768129699073, −7.38963929726500432671103897062, −5.32976813596212009628341324257, −4.25184226383277670654100093396, −3.53092108239114739520892682373, −2.53689775750779063677075983797, −1.02567628181629114810108559149, −0.16993579673048721963410344671,
1.40066779940805387866103149014, 2.45293969319022002090771763996, 4.42576954039789068254073938924, 5.82167690378175226083929219839, 6.42944725765693098490762322001, 7.46554379039026171309318262554, 8.290517533995335334833952859953, 9.354019541161022945943651520016, 11.46977052107089203362493994325, 12.62105537901448215403136932022
