L(s) = 1 | + (4.20e4 + 4.20e4i)2-s + (2.23e7 − 2.23e7i)3-s − 7.62e8i·4-s + 1.87e12·6-s + (2.39e13 + 2.39e13i)7-s + (2.12e14 − 2.12e14i)8-s + 8.55e14i·9-s − 3.17e16·11-s + (−1.70e16 − 1.70e16i)12-s + (5.32e17 − 5.32e17i)13-s + 2.01e18i·14-s + 1.45e19·16-s + (−4.10e19 − 4.10e19i)17-s + (−3.59e19 + 3.59e19i)18-s + 2.22e20i·19-s + ⋯ |
L(s) = 1 | + (0.641 + 0.641i)2-s + (0.518 − 0.518i)3-s − 0.177i·4-s + 0.665·6-s + (0.721 + 0.721i)7-s + (0.755 − 0.755i)8-s + 0.461i·9-s − 0.690·11-s + (−0.0920 − 0.0920i)12-s + (0.800 − 0.800i)13-s + 0.925i·14-s + 0.791·16-s + (−0.842 − 0.842i)17-s + (−0.295 + 0.295i)18-s + 0.769i·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
\(L(\frac{33}{2})\) |
\(\approx\) |
\(4.711880548\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.711880548\) |
\(L(17)\) |
|
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 | \( 1 + (-4.20e4 - 4.20e4i)T + 4.29e9iT^{2} \) |
| 3 | \( 1 + (-2.23e7 + 2.23e7i)T - 1.85e15iT^{2} \) |
| 7 | \( 1 + (-2.39e13 - 2.39e13i)T + 1.10e27iT^{2} \) |
| 11 | \( 1 + 3.17e16T + 2.11e33T^{2} \) |
| 13 | \( 1 + (-5.32e17 + 5.32e17i)T - 4.42e35iT^{2} \) |
| 17 | \( 1 + (4.10e19 + 4.10e19i)T + 2.36e39iT^{2} \) |
| 19 | \( 1 - 2.22e20iT - 8.31e40T^{2} \) |
| 23 | \( 1 + (-6.22e21 + 6.22e21i)T - 3.76e43iT^{2} \) |
| 29 | \( 1 + 3.25e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 - 1.21e22T + 5.29e47T^{2} \) |
| 37 | \( 1 + (-1.53e25 - 1.53e25i)T + 1.52e50iT^{2} \) |
| 41 | \( 1 + 2.70e25T + 4.06e51T^{2} \) |
| 43 | \( 1 + (9.54e25 - 9.54e25i)T - 1.86e52iT^{2} \) |
| 47 | \( 1 + (2.53e26 + 2.53e26i)T + 3.21e53iT^{2} \) |
| 53 | \( 1 + (-1.49e27 + 1.49e27i)T - 1.50e55iT^{2} \) |
| 59 | \( 1 - 2.48e28iT - 4.64e56T^{2} \) |
| 61 | \( 1 - 3.17e28T + 1.35e57T^{2} \) |
| 67 | \( 1 + (-3.16e27 - 3.16e27i)T + 2.71e58iT^{2} \) |
| 71 | \( 1 - 4.69e29T + 1.73e59T^{2} \) |
| 73 | \( 1 + (-3.04e29 + 3.04e29i)T - 4.22e59iT^{2} \) |
| 79 | \( 1 + 1.79e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 + (-4.68e29 + 4.68e29i)T - 2.57e61iT^{2} \) |
| 89 | \( 1 + 2.97e31iT - 2.40e62T^{2} \) |
| 97 | \( 1 + (-1.89e31 - 1.89e31i)T + 3.77e63iT^{2} \) |
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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
−11.36098412834771881250307397076, −10.19158718344525720144372780902, −8.524317859271610060600661629433, −7.77409228222579182421421926842, −6.50209837588593035249778342222, −5.37739446692928571557212574293, −4.59533875392113017770499534751, −2.88305374935718369506285138136, −1.87983930546614055971889057740, −0.70196236767059640596260524915,
1.09821689887979683369222396953, 2.22459092512705401745147678774, 3.43125568407296750769441312265, 4.11589540101123488397591277050, 5.06790242361337104487852341173, 6.88388386857469092308846549620, 8.171107189880214122859650285257, 9.161816363556000970989491201984, 10.77436976200158051640728061209, 11.30907453368905270979398624213