| L(s) = 1 | + (1.28 + 0.587i)2-s + (−0.271 + 0.924i)3-s + (1.30 + 1.51i)4-s + (−2.81 − 4.13i)5-s + (−0.892 + 1.02i)6-s + (1.64 − 11.4i)7-s + (0.796 + 2.71i)8-s + (6.79 + 4.36i)9-s + (−1.19 − 6.96i)10-s + (12.0 − 5.49i)11-s + (−1.75 + 0.800i)12-s + (−2.01 + 0.289i)13-s + (8.81 − 13.7i)14-s + (4.58 − 1.48i)15-s + (−0.569 + 3.95i)16-s + (1.75 − 2.02i)17-s + ⋯ |
| L(s) = 1 | + (0.643 + 0.293i)2-s + (−0.0904 + 0.308i)3-s + (0.327 + 0.377i)4-s + (−0.563 − 0.826i)5-s + (−0.148 + 0.171i)6-s + (0.234 − 1.63i)7-s + (0.0996 + 0.339i)8-s + (0.754 + 0.484i)9-s + (−0.119 − 0.696i)10-s + (1.09 − 0.499i)11-s + (−0.146 + 0.0666i)12-s + (−0.155 + 0.0222i)13-s + (0.629 − 0.980i)14-s + (0.305 − 0.0988i)15-s + (−0.0355 + 0.247i)16-s + (0.103 − 0.119i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.14694 - 0.387773i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.14694 - 0.387773i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.28 - 0.587i)T \) |
| 5 | \( 1 + (2.81 + 4.13i)T \) |
| 23 | \( 1 + (14.0 - 18.2i)T \) |
| good | 3 | \( 1 + (0.271 - 0.924i)T + (-7.57 - 4.86i)T^{2} \) |
| 7 | \( 1 + (-1.64 + 11.4i)T + (-47.0 - 13.8i)T^{2} \) |
| 11 | \( 1 + (-12.0 + 5.49i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (2.01 - 0.289i)T + (162. - 47.6i)T^{2} \) |
| 17 | \( 1 + (-1.75 + 2.02i)T + (-41.1 - 286. i)T^{2} \) |
| 19 | \( 1 + (-15.6 + 13.5i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (1.93 - 2.23i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (-11.8 + 3.48i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (51.5 + 33.1i)T + (568. + 1.24e3i)T^{2} \) |
| 41 | \( 1 + (-49.8 + 32.0i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (-13.5 - 3.97i)T + (1.55e3 + 9.99e2i)T^{2} \) |
| 47 | \( 1 - 0.939iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-3.93 + 27.3i)T + (-2.69e3 - 791. i)T^{2} \) |
| 59 | \( 1 + (-7.29 - 50.7i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (-17.4 - 59.4i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (35.0 - 76.7i)T + (-2.93e3 - 3.39e3i)T^{2} \) |
| 71 | \( 1 + (50.3 - 110. i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (-11.4 + 9.93i)T + (758. - 5.27e3i)T^{2} \) |
| 79 | \( 1 + (113. - 16.3i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (-41.8 - 26.8i)T + (2.86e3 + 6.26e3i)T^{2} \) |
| 89 | \( 1 + (-2.06 + 7.04i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (1.69 - 1.08i)T + (3.90e3 - 8.55e3i)T^{2} \) |
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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
−11.90723627075646274858333676766, −11.16287008104168608313606385587, −10.09887422606218318420342550813, −8.927062049403039741932887965343, −7.57489395101631004889687081164, −7.11129185928173219532336954037, −5.43150771448411757497829154585, −4.29200317863581889399854363704, −3.79857989693187481075322252505, −1.14484804601799005026015060441,
1.84443322184548962384219935171, 3.24570810666159867887355513190, 4.49416747642506802419399790354, 5.97452228230101137217272452145, 6.72483989121081771731088783646, 7.913655151178078843038597392719, 9.254261998996838215053481496937, 10.20194723837666582762269564567, 11.53154157356709188373493149586, 12.12258067251535561671387701162
