| L(s) = 1 | + (−1.00 + 0.619i)2-s + (0.241 + 0.0451i)3-s + (−0.273 + 0.549i)4-s + (−0.624 − 2.19i)5-s + (−0.270 + 0.104i)6-s + 3.32·7-s + (−0.283 − 3.06i)8-s + (−2.74 − 1.06i)9-s + (1.98 + 1.80i)10-s + (−2.74 − 1.06i)11-s + (−0.0909 + 0.120i)12-s + (−5.96 + 2.30i)13-s + (−3.32 + 2.05i)14-s + (−0.0517 − 0.558i)15-s + (1.44 + 1.91i)16-s + (0.297 + 3.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.438i)2-s + (0.139 + 0.0260i)3-s + (−0.136 + 0.274i)4-s + (−0.279 − 0.981i)5-s + (−0.110 + 0.0427i)6-s + 1.25·7-s + (−0.100 − 1.08i)8-s + (−0.913 − 0.353i)9-s + (0.627 + 0.572i)10-s + (−0.827 − 0.320i)11-s + (−0.0262 + 0.0347i)12-s + (−1.65 + 0.640i)13-s + (−0.888 + 0.550i)14-s + (−0.0133 − 0.144i)15-s + (0.360 + 0.477i)16-s + (0.0722 + 0.779i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.223242 - 0.314817i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.223242 - 0.314817i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 409 | \( 1 + (2.47 + 20.0i)T \) |
| good | 2 | \( 1 + (1.00 - 0.619i)T + (0.891 - 1.79i)T^{2} \) |
| 3 | \( 1 + (-0.241 - 0.0451i)T + (2.79 + 1.08i)T^{2} \) |
| 5 | \( 1 + (0.624 + 2.19i)T + (-4.25 + 2.63i)T^{2} \) |
| 7 | \( 1 - 3.32T + 7T^{2} \) |
| 11 | \( 1 + (2.74 + 1.06i)T + (8.12 + 7.41i)T^{2} \) |
| 13 | \( 1 + (5.96 - 2.30i)T + (9.60 - 8.75i)T^{2} \) |
| 17 | \( 1 + (-0.297 - 3.21i)T + (-16.7 + 3.12i)T^{2} \) |
| 19 | \( 1 + (3.13 + 1.93i)T + (8.46 + 17.0i)T^{2} \) |
| 23 | \( 1 + (0.715 + 7.71i)T + (-22.6 + 4.22i)T^{2} \) |
| 29 | \( 1 + (7.22 + 6.58i)T + (2.67 + 28.8i)T^{2} \) |
| 31 | \( 1 - 6.42T + 31T^{2} \) |
| 37 | \( 1 + (-2.68 + 3.55i)T + (-10.1 - 35.5i)T^{2} \) |
| 41 | \( 1 + (2.32 - 2.12i)T + (3.78 - 40.8i)T^{2} \) |
| 43 | \( 1 + (2.43 - 1.50i)T + (19.1 - 38.4i)T^{2} \) |
| 47 | \( 1 + (5.01 - 6.64i)T + (-12.8 - 45.2i)T^{2} \) |
| 53 | \( 1 - 9.00T + 53T^{2} \) |
| 59 | \( 1 + (-3.73 + 7.49i)T + (-35.5 - 47.0i)T^{2} \) |
| 61 | \( 1 + (7.46 - 1.39i)T + (56.8 - 22.0i)T^{2} \) |
| 67 | \( 1 + (1.73 + 0.673i)T + (49.5 + 45.1i)T^{2} \) |
| 71 | \( 1 + (-2.26 + 7.95i)T + (-60.3 - 37.3i)T^{2} \) |
| 73 | \( 1 + (2.53 + 5.09i)T + (-43.9 + 58.2i)T^{2} \) |
| 79 | \( 1 + (-1.66 - 5.84i)T + (-67.1 + 41.5i)T^{2} \) |
| 83 | \( 1 + (-0.168 - 1.81i)T + (-81.5 + 15.2i)T^{2} \) |
| 89 | \( 1 + (-6.00 - 12.0i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (-3.00 - 10.5i)T + (-82.4 + 51.0i)T^{2} \) |
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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
−10.93316870676904133531281949160, −9.750698199863313264203605113190, −8.824022042197943981898927178286, −8.204576816658631474693085084893, −7.76453087678658078376942873243, −6.37223659087675157362209145941, −4.95227522389253096531699039455, −4.27022491368076898977615551332, −2.42315686624287784349615402522, −0.29247686232028239471896809040,
2.02358942356593737082890005091, 2.95294258232952838326044913330, 4.96892180243867937292309698072, 5.47093898332723629431618495889, 7.33886154792692760472440877743, 7.83991577520632308870738877409, 8.771776746023221999397074826003, 9.975685012649347383789536007428, 10.50676853490753204737695036562, 11.39367671242995065818338322253
