| L(s) = 1 | + (−1.08 + 2.37i)3-s + (−0.654 + 0.755i)5-s + (0.0544 − 0.0349i)7-s + (−2.51 − 2.90i)9-s + (−0.824 + 5.73i)11-s + (−3.15 − 2.02i)13-s + (−1.08 − 2.37i)15-s + (−1.70 − 0.501i)17-s + (3.27 − 0.960i)19-s + (0.0241 + 0.167i)21-s + (−4.09 − 2.49i)23-s + (−0.142 − 0.989i)25-s + (2.11 − 0.621i)27-s + (−1.26 − 0.371i)29-s + (−1.01 − 2.23i)31-s + ⋯ |
| L(s) = 1 | + (−0.627 + 1.37i)3-s + (−0.292 + 0.337i)5-s + (0.0205 − 0.0132i)7-s + (−0.838 − 0.968i)9-s + (−0.248 + 1.72i)11-s + (−0.874 − 0.561i)13-s + (−0.280 − 0.614i)15-s + (−0.414 − 0.121i)17-s + (0.750 − 0.220i)19-s + (0.00525 + 0.0365i)21-s + (−0.854 − 0.519i)23-s + (−0.0284 − 0.197i)25-s + (0.407 − 0.119i)27-s + (−0.234 − 0.0689i)29-s + (−0.183 − 0.400i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0760430 - 0.552472i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0760430 - 0.552472i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (4.09 + 2.49i)T \) |
| good | 3 | \( 1 + (1.08 - 2.37i)T + (-1.96 - 2.26i)T^{2} \) |
| 7 | \( 1 + (-0.0544 + 0.0349i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.824 - 5.73i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (3.15 + 2.02i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (1.70 + 0.501i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-3.27 + 0.960i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (1.26 + 0.371i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (1.01 + 2.23i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (1.23 + 1.42i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (0.0704 - 0.0812i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (0.298 - 0.653i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 0.878T + 47T^{2} \) |
| 53 | \( 1 + (7.32 - 4.71i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-12.8 - 8.27i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-0.555 - 1.21i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.53 - 10.6i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.57 - 10.9i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (4.82 - 1.41i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (6.43 + 4.13i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-8.70 - 10.0i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.981 + 2.14i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-4.21 + 4.86i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.43483604946164158030272162026, −10.42302128844435995897942052356, −9.970263254324160281776587594183, −9.256520522926861721296406313705, −7.80268902923116609484328224653, −6.97867479147699527073010133873, −5.60984074272920761715822870283, −4.74357440703907152495576327854, −4.03582700587593977636169614992, −2.53973384707563534937400676898,
0.35925108431913601923553835849, 1.84845189212806232157372062991, 3.44458904425183650933808770023, 5.08958169956476142291433539455, 5.95769632650133527614740746588, 6.84000064960477398154747127395, 7.77544090457859237534979139970, 8.462364983111926255053841869164, 9.627981993639191077262953162557, 10.95416067809024371903982299989
