L(s) = 1 | + (−0.592 − 1.62i)3-s + (1.63 − 0.943i)7-s + (−2.29 + 1.92i)9-s + (−6.08 − 2.21i)13-s + (−0.5 − 4.33i)19-s + (−2.50 − 2.10i)21-s + (−4.69 − 1.71i)25-s + (4.5 + 2.59i)27-s + (4.66 − 2.69i)31-s − 3.20·37-s + 11.2i·39-s + (−12.9 − 2.27i)43-s + (−1.71 + 2.97i)49-s + (−6.75 + 3.37i)57-s + (−0.762 − 4.32i)61-s + ⋯ |
L(s) = 1 | + (−0.342 − 0.939i)3-s + (0.617 − 0.356i)7-s + (−0.766 + 0.642i)9-s + (−1.68 − 0.614i)13-s + (−0.114 − 0.993i)19-s + (−0.546 − 0.458i)21-s + (−0.939 − 0.342i)25-s + (0.866 + 0.499i)27-s + (0.838 − 0.484i)31-s − 0.527·37-s + 1.79i·39-s + (−1.96 − 0.346i)43-s + (−0.245 + 0.425i)49-s + (−0.894 + 0.447i)57-s + (−0.0976 − 0.553i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0455479 - 0.699137i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0455479 - 0.699137i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.592 + 1.62i)T \) |
| 19 | \( 1 + (0.5 + 4.33i)T \) |
good | 5 | \( 1 + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.63 + 0.943i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (6.08 + 2.21i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.66 + 2.69i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.20T + 37T^{2} \) |
| 41 | \( 1 + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (12.9 + 2.27i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.762 + 4.32i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.42 - 1.69i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (14.4 - 5.26i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (5.09 + 14.0i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-14.5 - 12.2i)T + (16.8 + 95.5i)T^{2} \) |
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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
−9.838455636403342876981847841219, −8.638106879942035239595369428270, −7.79343663371102897794041158302, −7.26773525914116564781467947232, −6.35634755325490804292787892146, −5.25634109765324018923881494419, −4.59118378753662316257461448643, −2.91336709552213079249836290300, −1.88871243830551234459117170696, −0.32589764396672231637462322538,
1.96513582016621737070182723937, 3.29135045865570922281486033021, 4.47319995445359238366800532126, 5.07831340279622912027278902826, 5.99246028745763803769032047268, 7.08162784268925965696350957488, 8.118940482518010667523702040609, 8.913420008215262274449393610519, 9.943128470987361861770344572843, 10.16352510462839174249652829619