L(s) = 1 | + (−0.654 − 0.755i)3-s + (−1.78 + 0.713i)7-s + (−0.142 + 0.989i)9-s + (−1.84 − 0.176i)13-s + (−1.21 − 0.486i)19-s + (1.70 + 0.879i)21-s + (0.415 + 0.909i)25-s + (0.841 − 0.540i)27-s + (−0.0947 + 0.00904i)31-s + (−0.415 + 0.719i)37-s + (1.07 + 1.51i)39-s + (−1.11 + 0.326i)43-s + (1.94 − 1.85i)49-s + (0.428 + 1.23i)57-s + (−0.0311 − 0.653i)61-s + ⋯ |
L(s) = 1 | + (−0.654 − 0.755i)3-s + (−1.78 + 0.713i)7-s + (−0.142 + 0.989i)9-s + (−1.84 − 0.176i)13-s + (−1.21 − 0.486i)19-s + (1.70 + 0.879i)21-s + (0.415 + 0.909i)25-s + (0.841 − 0.540i)27-s + (−0.0947 + 0.00904i)31-s + (−0.415 + 0.719i)37-s + (1.07 + 1.51i)39-s + (−1.11 + 0.326i)43-s + (1.94 − 1.85i)49-s + (0.428 + 1.23i)57-s + (−0.0311 − 0.653i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07085363766\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07085363766\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.654 + 0.755i)T \) |
| 67 | \( 1 + (-0.580 + 0.814i)T \) |
good | 5 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 7 | \( 1 + (1.78 - 0.713i)T + (0.723 - 0.690i)T^{2} \) |
| 11 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 13 | \( 1 + (1.84 + 0.176i)T + (0.981 + 0.189i)T^{2} \) |
| 17 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 19 | \( 1 + (1.21 + 0.486i)T + (0.723 + 0.690i)T^{2} \) |
| 23 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.0947 - 0.00904i)T + (0.981 - 0.189i)T^{2} \) |
| 37 | \( 1 + (0.415 - 0.719i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 43 | \( 1 + (1.11 - 0.326i)T + (0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 53 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 59 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 61 | \( 1 + (0.0311 + 0.653i)T + (-0.995 + 0.0950i)T^{2} \) |
| 71 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 73 | \( 1 + (0.0845 + 1.77i)T + (-0.995 + 0.0950i)T^{2} \) |
| 79 | \( 1 + (1.15 - 1.62i)T + (-0.327 - 0.945i)T^{2} \) |
| 83 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 89 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 97 | \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.78333257166456479703168808552, −9.923268018952180336787588168805, −9.266260193570025961979737140673, −8.182708334998171153822149094449, −6.99892634373789375070053269136, −6.65121482210047489679564943262, −5.64975353313503496422065975309, −4.79031317376222176044262898515, −3.13038467398843644213422975509, −2.23035480279391141555863332827,
0.07145287117382331491844641735, 2.68429372184684496228467490814, 3.80943809978851047755496325403, 4.59556424378235864888375588361, 5.75628197610029297159132000856, 6.65307353239513138322182539749, 7.19807796642669736926299128956, 8.678696166762429505098501282922, 9.642259803914182893985597164102, 10.10705407753663906938624361824