| L(s) = 1 | + (−1.79 − 1.22i)2-s + (1.34 + 1.24i)3-s + (0.991 + 2.52i)4-s + (1.96 − 0.607i)5-s + (−0.884 − 3.87i)6-s + (1.60 + 2.10i)7-s + (0.344 − 1.51i)8-s + (0.0264 + 0.353i)9-s + (−4.27 − 1.31i)10-s + (0.146 − 1.95i)11-s + (−1.81 + 4.62i)12-s + (0.900 − 0.433i)13-s + (−0.300 − 5.73i)14-s + (3.40 + 1.63i)15-s + (1.51 − 1.40i)16-s + (4.13 − 0.623i)17-s + ⋯ |
| L(s) = 1 | + (−1.26 − 0.864i)2-s + (0.775 + 0.719i)3-s + (0.495 + 1.26i)4-s + (0.880 − 0.271i)5-s + (−0.361 − 1.58i)6-s + (0.605 + 0.795i)7-s + (0.121 − 0.534i)8-s + (0.00881 + 0.117i)9-s + (−1.35 − 0.417i)10-s + (0.0441 − 0.588i)11-s + (−0.524 + 1.33i)12-s + (0.249 − 0.120i)13-s + (−0.0802 − 1.53i)14-s + (0.877 + 0.422i)15-s + (0.377 − 0.350i)16-s + (1.00 − 0.151i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25970 - 0.260778i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25970 - 0.260778i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.60 - 2.10i)T \) |
| 13 | \( 1 + (-0.900 + 0.433i)T \) |
| good | 2 | \( 1 + (1.79 + 1.22i)T + (0.730 + 1.86i)T^{2} \) |
| 3 | \( 1 + (-1.34 - 1.24i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-1.96 + 0.607i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.146 + 1.95i)T + (-10.8 - 1.63i)T^{2} \) |
| 17 | \( 1 + (-4.13 + 0.623i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (2.88 + 4.99i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.19 - 0.180i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-4.84 - 6.07i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (5.35 - 9.27i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.147 + 0.376i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-2.35 + 10.3i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.267 - 1.17i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-10.7 - 7.35i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (-1.56 - 3.97i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (8.61 + 2.65i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (2.15 - 5.48i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-3.49 + 6.04i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.28 + 2.85i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.0413 + 0.0281i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (6.04 + 10.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.62 + 4.15i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (0.575 + 7.68i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 - 0.774T + 97T^{2} \) |
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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
−10.48778003554024948074961411807, −9.426623113118294858157075473852, −8.835873824915910867513781011928, −8.714518477225643525193587158358, −7.45264975674344763035718226221, −5.89792791937306398401492309011, −4.94323981827083382135407936106, −3.32724133048316571458324085303, −2.52127641060666391959593747148, −1.29187555643232272079814916241,
1.28940123569469301167854058962, 2.23505267824437288384961046115, 4.01849059405499945342473475400, 5.67729512649409452941842402370, 6.51873319719948989919435283434, 7.46718010199349945745458964834, 7.917670250644372188032065428938, 8.601473580897623987779602341183, 9.804773016483914455285868781779, 10.08139656442698907533087791477
