L(s) = 1 | + (−7.09e3 − 7.09e3i)2-s + (−4.70e7 + 4.70e7i)3-s − 4.19e9i·4-s + (−3.87e10 − 1.47e11i)5-s + 6.67e11·6-s + (−3.04e13 − 3.04e13i)7-s + (−6.02e13 + 6.02e13i)8-s − 2.57e15i·9-s + (−7.72e14 + 1.32e15i)10-s + 3.11e16·11-s + (1.97e17 + 1.97e17i)12-s + (8.11e17 − 8.11e17i)13-s + 4.32e17i·14-s + (8.76e18 + 5.12e18i)15-s − 1.71e19·16-s + (−5.99e19 − 5.99e19i)17-s + ⋯ |
L(s) = 1 | + (−0.108 − 0.108i)2-s + (−1.09 + 1.09i)3-s − 0.976i·4-s + (−0.253 − 0.967i)5-s + 0.236·6-s + (−0.915 − 0.915i)7-s + (−0.214 + 0.214i)8-s − 1.38i·9-s + (−0.0772 + 0.132i)10-s + 0.678·11-s + (1.06 + 1.06i)12-s + (1.21 − 1.21i)13-s + 0.198i·14-s + (1.33 + 0.779i)15-s − 0.930·16-s + (−1.23 − 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.724 - 0.689i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(0.3905142101\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3905142101\) |
\(L(17)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (3.87e10 + 1.47e11i)T \) |
good | 2 | \( 1 + (7.09e3 + 7.09e3i)T + 4.29e9iT^{2} \) |
| 3 | \( 1 + (4.70e7 - 4.70e7i)T - 1.85e15iT^{2} \) |
| 7 | \( 1 + (3.04e13 + 3.04e13i)T + 1.10e27iT^{2} \) |
| 11 | \( 1 - 3.11e16T + 2.11e33T^{2} \) |
| 13 | \( 1 + (-8.11e17 + 8.11e17i)T - 4.42e35iT^{2} \) |
| 17 | \( 1 + (5.99e19 + 5.99e19i)T + 2.36e39iT^{2} \) |
| 19 | \( 1 - 3.24e19iT - 8.31e40T^{2} \) |
| 23 | \( 1 + (-7.07e20 + 7.07e20i)T - 3.76e43iT^{2} \) |
| 29 | \( 1 + 1.29e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 - 1.94e23T + 5.29e47T^{2} \) |
| 37 | \( 1 + (1.30e25 + 1.30e25i)T + 1.52e50iT^{2} \) |
| 41 | \( 1 - 1.97e25T + 4.06e51T^{2} \) |
| 43 | \( 1 + (1.49e26 - 1.49e26i)T - 1.86e52iT^{2} \) |
| 47 | \( 1 + (1.46e25 + 1.46e25i)T + 3.21e53iT^{2} \) |
| 53 | \( 1 + (-1.61e26 + 1.61e26i)T - 1.50e55iT^{2} \) |
| 59 | \( 1 - 1.65e28iT - 4.64e56T^{2} \) |
| 61 | \( 1 - 3.79e28T + 1.35e57T^{2} \) |
| 67 | \( 1 + (5.62e27 + 5.62e27i)T + 2.71e58iT^{2} \) |
| 71 | \( 1 - 2.56e29T + 1.73e59T^{2} \) |
| 73 | \( 1 + (-2.59e28 + 2.59e28i)T - 4.22e59iT^{2} \) |
| 79 | \( 1 - 6.46e29iT - 5.29e60T^{2} \) |
| 83 | \( 1 + (-5.85e30 + 5.85e30i)T - 2.57e61iT^{2} \) |
| 89 | \( 1 - 1.03e31iT - 2.40e62T^{2} \) |
| 97 | \( 1 + (-4.86e31 - 4.86e31i)T + 3.77e63iT^{2} \) |
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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
−15.60045934119855186567837526130, −13.37792195221756647752007696070, −11.41066244437791222398129145112, −10.31597838112529023975336026577, −9.183742925670322821644030654431, −6.35808296419895822607684435518, −5.10769369222516075374482509527, −3.91011796903652939556086215076, −0.857489745379136954736464064284, −0.19851629527802802840232879137,
1.94641505677362411333241846732, 3.64772876669741944271599690016, 6.41454359253828709871233862550, 6.71707311773534381713326419044, 8.691425502513242175911901940777, 11.23324174653847646758129196113, 12.13481523668164772160826436657, 13.36894772825381475390113188397, 15.72345060658732072669706659613, 17.12131302765545473759569285227