L(s) = 1 | + (−0.951 + 0.309i)2-s + (1.52 + 0.814i)3-s + (0.809 − 0.587i)4-s + (−0.866 + 0.5i)5-s + (−1.70 − 0.302i)6-s + (−0.239 − 2.27i)7-s + (−0.587 + 0.809i)8-s + (1.67 + 2.49i)9-s + (0.669 − 0.743i)10-s + (4.80 + 2.13i)11-s + (1.71 − 0.239i)12-s + (0.102 + 0.482i)13-s + (0.932 + 2.09i)14-s + (−1.73 + 0.0585i)15-s + (0.309 − 0.951i)16-s + (0.224 − 0.100i)17-s + ⋯ |
L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.882 + 0.470i)3-s + (0.404 − 0.293i)4-s + (−0.387 + 0.223i)5-s + (−0.696 − 0.123i)6-s + (−0.0905 − 0.861i)7-s + (−0.207 + 0.286i)8-s + (0.557 + 0.830i)9-s + (0.211 − 0.235i)10-s + (1.44 + 0.644i)11-s + (0.495 − 0.0690i)12-s + (0.0284 + 0.133i)13-s + (0.249 + 0.559i)14-s + (−0.446 + 0.0151i)15-s + (0.0772 − 0.237i)16-s + (0.0545 − 0.0242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40975 + 0.718259i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40975 + 0.718259i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (-1.52 - 0.814i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + (-4.86 + 2.71i)T \) |
good | 7 | \( 1 + (0.239 + 2.27i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-4.80 - 2.13i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.102 - 0.482i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-0.224 + 0.100i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (2.25 + 0.479i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-0.241 - 0.175i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.303 - 0.935i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-4.06 - 2.34i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.06 - 6.36i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-0.238 + 1.12i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-3.93 - 1.27i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.504 + 4.79i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (10.8 - 9.78i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 11.6iT - 61T^{2} \) |
| 67 | \( 1 + (4.94 + 8.55i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.46 - 0.679i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (0.845 - 1.89i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (1.47 + 3.32i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.758 + 0.842i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (11.3 - 8.27i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (2.96 - 2.15i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.983453110576686871656382604116, −9.359361410373730552939622804881, −8.609643534413126336150014951877, −7.68880976831289985442656866208, −7.08700286074493805130343923106, −6.22133407373572102624832942388, −4.50264971324539656232206588238, −3.99262813816738946611230939368, −2.72903637056953270454820653814, −1.32106052330890783514468828884,
1.01379580266264466640656589794, 2.27195455094275422155218582744, 3.32749078437722407841354202395, 4.25688770196499185928986290225, 5.94867144901460293477019760069, 6.66212735147096583732257277293, 7.68551453597650992728496194657, 8.461155613099878906306973876940, 9.022353766151965765506750286047, 9.536485206978349225027896048589