| L(s) = 1 | + (−1.07e5 + 6.21e4i)2-s + (−3.45e7 − 2.56e7i)3-s + (5.58e9 − 9.67e9i)4-s + (4.70e10 − 2.71e10i)5-s + (5.31e12 + 6.13e11i)6-s + (−5.64e12 + 3.27e13i)7-s + (−0.0937 + 8.55e14i)8-s + (5.37e14 + 1.77e15i)9-s + (−3.37e15 + 5.85e15i)10-s + (7.71e16 + 4.45e16i)11-s + (−4.41e17 + 1.91e17i)12-s − 7.94e17·13-s + (−1.42e18 − 3.87e18i)14-s + (−2.32e18 − 2.67e17i)15-s + (−2.91e19 − 5.05e19i)16-s + (2.93e19 + 1.69e19i)17-s + ⋯ |
| L(s) = 1 | + (−1.64 + 0.948i)2-s + (−0.803 − 0.595i)3-s + (1.30 − 2.25i)4-s + (0.308 − 0.178i)5-s + (1.88 + 0.217i)6-s + (−0.169 + 0.985i)7-s + 3.03i·8-s + (0.289 + 0.957i)9-s + (−0.337 + 0.585i)10-s + (1.67 + 0.969i)11-s + (−2.38 + 1.03i)12-s − 1.19·13-s + (−0.656 − 1.78i)14-s + (−0.353 − 0.0407i)15-s + (−1.58 − 2.74i)16-s + (0.603 + 0.348i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(0.2205920261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2205920261\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.45e7 + 2.56e7i)T \) |
| 7 | \( 1 + (5.64e12 - 3.27e13i)T \) |
| good | 2 | \( 1 + (1.07e5 - 6.21e4i)T + (2.14e9 - 3.71e9i)T^{2} \) |
| 5 | \( 1 + (-4.70e10 + 2.71e10i)T + (1.16e22 - 2.01e22i)T^{2} \) |
| 11 | \( 1 + (-7.71e16 - 4.45e16i)T + (1.05e33 + 1.82e33i)T^{2} \) |
| 13 | \( 1 + 7.94e17T + 4.42e35T^{2} \) |
| 17 | \( 1 + (-2.93e19 - 1.69e19i)T + (1.18e39 + 2.05e39i)T^{2} \) |
| 19 | \( 1 + (8.82e19 + 1.52e20i)T + (-4.15e40 + 7.20e40i)T^{2} \) |
| 23 | \( 1 + (3.84e21 - 2.22e21i)T + (1.88e43 - 3.25e43i)T^{2} \) |
| 29 | \( 1 + 1.60e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 + (1.34e23 - 2.32e23i)T + (-2.64e47 - 4.58e47i)T^{2} \) |
| 37 | \( 1 + (7.18e24 + 1.24e25i)T + (-7.61e49 + 1.31e50i)T^{2} \) |
| 41 | \( 1 - 9.83e25iT - 4.06e51T^{2} \) |
| 43 | \( 1 + 9.19e25T + 1.86e52T^{2} \) |
| 47 | \( 1 + (-1.47e26 + 8.51e25i)T + (1.60e53 - 2.78e53i)T^{2} \) |
| 53 | \( 1 + (3.89e27 + 2.24e27i)T + (7.51e54 + 1.30e55i)T^{2} \) |
| 59 | \( 1 + (-2.66e26 - 1.53e26i)T + (2.32e56 + 4.02e56i)T^{2} \) |
| 61 | \( 1 + (2.79e28 + 4.84e28i)T + (-6.75e56 + 1.16e57i)T^{2} \) |
| 67 | \( 1 + (-1.99e28 + 3.45e28i)T + (-1.35e58 - 2.35e58i)T^{2} \) |
| 71 | \( 1 + 6.55e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + (3.41e29 - 5.91e29i)T + (-2.11e59 - 3.66e59i)T^{2} \) |
| 79 | \( 1 + (4.19e28 + 7.26e28i)T + (-2.64e60 + 4.58e60i)T^{2} \) |
| 83 | \( 1 - 7.23e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + (1.20e31 - 6.96e30i)T + (1.20e62 - 2.07e62i)T^{2} \) |
| 97 | \( 1 - 6.65e31T + 3.77e63T^{2} \) |
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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
−11.45475306364329619370584361166, −9.873149347873675767549683667224, −9.277604896257719223478318420708, −7.88281044249269439109473457110, −6.85664073533609625357482525440, −6.08879744653101817558821324933, −5.03133663313238123281000713558, −2.03124874368343780176946402062, −1.48640536786542359603448040896, −0.13233633959119931121537046929,
0.67616087335380247644592014130, 1.58441580607853465831455330864, 3.20950939186916434177523085726, 4.13468445102567660624734217299, 6.29076024120616763455380690241, 7.31249751480319460541006009659, 8.836515241981622340086713706236, 9.889759329851282830416406851120, 10.42327499807718098892552582423, 11.59160878610322072012863921573
