L(s) = 1 | + (−2.43e4 + 2.43e4i)2-s + (2.67e7 + 2.67e7i)3-s + 3.10e9i·4-s − 1.30e12·6-s + (−4.33e13 + 4.33e13i)7-s + (−1.80e14 − 1.80e14i)8-s − 4.16e14i·9-s + 2.50e16·11-s + (−8.33e16 + 8.33e16i)12-s + (7.52e17 + 7.52e17i)13-s − 2.11e18i·14-s − 4.57e18·16-s + (3.85e19 − 3.85e19i)17-s + (1.01e19 + 1.01e19i)18-s + 1.17e20i·19-s + ⋯ |
L(s) = 1 | + (−0.371 + 0.371i)2-s + (0.622 + 0.622i)3-s + 0.723i·4-s − 0.462·6-s + (−1.30 + 1.30i)7-s + (−0.640 − 0.640i)8-s − 0.225i·9-s + 0.545·11-s + (−0.450 + 0.450i)12-s + (1.13 + 1.13i)13-s − 0.970i·14-s − 0.247·16-s + (0.791 − 0.791i)17-s + (0.0836 + 0.0836i)18-s + 0.405i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(1.486894220\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.486894220\) |
\(L(17)\) |
|
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 | \( 1 + (2.43e4 - 2.43e4i)T - 4.29e9iT^{2} \) |
| 3 | \( 1 + (-2.67e7 - 2.67e7i)T + 1.85e15iT^{2} \) |
| 7 | \( 1 + (4.33e13 - 4.33e13i)T - 1.10e27iT^{2} \) |
| 11 | \( 1 - 2.50e16T + 2.11e33T^{2} \) |
| 13 | \( 1 + (-7.52e17 - 7.52e17i)T + 4.42e35iT^{2} \) |
| 17 | \( 1 + (-3.85e19 + 3.85e19i)T - 2.36e39iT^{2} \) |
| 19 | \( 1 - 1.17e20iT - 8.31e40T^{2} \) |
| 23 | \( 1 + (3.24e21 + 3.24e21i)T + 3.76e43iT^{2} \) |
| 29 | \( 1 + 2.76e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 + 7.57e23T + 5.29e47T^{2} \) |
| 37 | \( 1 + (-6.38e24 + 6.38e24i)T - 1.52e50iT^{2} \) |
| 41 | \( 1 + 4.00e25T + 4.06e51T^{2} \) |
| 43 | \( 1 + (-2.44e24 - 2.44e24i)T + 1.86e52iT^{2} \) |
| 47 | \( 1 + (-7.35e26 + 7.35e26i)T - 3.21e53iT^{2} \) |
| 53 | \( 1 + (4.27e26 + 4.27e26i)T + 1.50e55iT^{2} \) |
| 59 | \( 1 + 1.79e28iT - 4.64e56T^{2} \) |
| 61 | \( 1 + 1.42e28T + 1.35e57T^{2} \) |
| 67 | \( 1 + (-1.58e29 + 1.58e29i)T - 2.71e58iT^{2} \) |
| 71 | \( 1 - 2.34e29T + 1.73e59T^{2} \) |
| 73 | \( 1 + (-1.18e29 - 1.18e29i)T + 4.22e59iT^{2} \) |
| 79 | \( 1 + 1.26e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 + (7.76e29 + 7.76e29i)T + 2.57e61iT^{2} \) |
| 89 | \( 1 - 4.78e30iT - 2.40e62T^{2} \) |
| 97 | \( 1 + (1.96e31 - 1.96e31i)T - 3.77e63iT^{2} \) |
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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
−11.75479164558686059235789233023, −9.658256928864380617289585084030, −9.183156847642349850549414762476, −8.380995795284224080948307937458, −6.76844236747039043304587388561, −5.95560848622283607058274633861, −3.93740775445052870353176092581, −3.37456159371074066851357859732, −2.27837474765996915036296291353, −0.35828884678756975001035001026,
0.913759327674928722892903881603, 1.39792013054657502836338962277, 2.90726325862829462171582167152, 3.78498442200310625234093085068, 5.65822344655387153967425622991, 6.70480759894145550262492696830, 7.85493009610969128362918352271, 9.088497224791412139477006186222, 10.23604293920114937185472684194, 10.88156329210619794502803961808