| L(s) = 1 | + (0.0904 − 0.187i)2-s + (−1.50 + 0.853i)3-s + (1.21 + 1.52i)4-s + (−4.15 − 1.28i)5-s + (0.0239 + 0.360i)6-s + (1.29 − 2.30i)7-s + (0.803 − 0.183i)8-s + (1.54 − 2.57i)9-s + (−0.616 + 0.664i)10-s + (2.20 − 0.165i)11-s + (−3.14 − 1.26i)12-s + (0.293 − 0.0220i)13-s + (−0.315 − 0.452i)14-s + (7.35 − 1.61i)15-s + (−0.832 + 3.64i)16-s + (1.14 + 2.91i)17-s + ⋯ |
| L(s) = 1 | + (0.0639 − 0.132i)2-s + (−0.870 + 0.492i)3-s + (0.609 + 0.764i)4-s + (−1.85 − 0.573i)5-s + (0.00975 + 0.147i)6-s + (0.491 − 0.870i)7-s + (0.284 − 0.0648i)8-s + (0.514 − 0.857i)9-s + (−0.194 + 0.210i)10-s + (0.665 − 0.0498i)11-s + (−0.907 − 0.365i)12-s + (0.0814 − 0.00610i)13-s + (−0.0842 − 0.120i)14-s + (1.89 − 0.416i)15-s + (−0.208 + 0.911i)16-s + (0.277 + 0.707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.979998 - 0.164105i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.979998 - 0.164105i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.50 - 0.853i)T \) |
| 7 | \( 1 + (-1.29 + 2.30i)T \) |
| good | 2 | \( 1 + (-0.0904 + 0.187i)T + (-1.24 - 1.56i)T^{2} \) |
| 5 | \( 1 + (4.15 + 1.28i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-2.20 + 0.165i)T + (10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (-0.293 + 0.0220i)T + (12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (-1.14 - 2.91i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-6.44 + 3.72i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.499 + 3.31i)T + (-21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-1.14 + 0.447i)T + (21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + 4.33iT - 31T^{2} \) |
| 37 | \( 1 + (-1.40 - 0.211i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (-6.53 + 6.06i)T + (3.06 - 40.8i)T^{2} \) |
| 43 | \( 1 + (2.89 + 2.68i)T + (3.21 + 42.8i)T^{2} \) |
| 47 | \( 1 + (-1.56 - 0.755i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (1.51 + 10.0i)T + (-50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (1.27 - 5.57i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (4.82 + 3.85i)T + (13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + (-8.46 + 6.75i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-6.63 - 0.496i)T + (72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + (0.558 - 7.45i)T + (-82.0 - 12.3i)T^{2} \) |
| 89 | \( 1 + (-7.50 - 5.11i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (4.59 + 2.65i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.17053061588899638393029491994, −10.71290140152263618100638351337, −9.216128365283852884430284613868, −8.089185176332961110341861191797, −7.45702833803094998294973755274, −6.62482888285663627358682310709, −4.95215744389701242017013533397, −4.06580355947961819305256867202, −3.53886537318179550382346724094, −0.863433482813690081115118056094,
1.23030346027172492694461414176, 3.03840056838544729406291367790, 4.57642944952034301924081863818, 5.55371853777205257007192725049, 6.55924797181970371666424387807, 7.45671053920872495387453395682, 7.946976468390381852729065785569, 9.481906203274454307727874185648, 10.72230036643167981160098517429, 11.35989218704353212970165053223
