L(s) = 1 | + (0.592 + 1.62i)3-s + (−2.47 − 4.28i)7-s + (−2.29 + 1.92i)9-s + (1.08 − 2.98i)13-s + (−0.5 − 4.33i)19-s + (5.50 − 6.55i)21-s + (−4.69 − 1.71i)25-s + (−4.5 − 2.59i)27-s + (4.66 − 2.69i)31-s − 11.7i·37-s + 5.49·39-s + (0.0957 − 0.543i)43-s + (−8.71 + 15.1i)49-s + (6.75 − 3.37i)57-s + (0.762 + 4.32i)61-s + ⋯ |
L(s) = 1 | + (0.342 + 0.939i)3-s + (−0.934 − 1.61i)7-s + (−0.766 + 0.642i)9-s + (0.300 − 0.826i)13-s + (−0.114 − 0.993i)19-s + (1.20 − 1.43i)21-s + (−0.939 − 0.342i)25-s + (−0.866 − 0.499i)27-s + (0.838 − 0.484i)31-s − 1.92i·37-s + 0.879·39-s + (0.0146 − 0.0828i)43-s + (−1.24 + 2.15i)49-s + (0.894 − 0.447i)57-s + (0.0976 + 0.553i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.883444 - 0.658209i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.883444 - 0.658209i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.47 + 4.28i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.08 + 2.98i)T + (-9.95 - 8.35i)T^{2} \) |
| 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 | \( 1 + 11.7iT - 37T^{2} \) |
| 41 | \( 1 + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.0957 + 0.543i)T + (-40.4 - 14.7i)T^{2} \) |
| 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 | \( 1 + (3.34 - 3.98i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−9.881512659784445312204333808307, −9.379735783572519491400951492973, −8.243072511662298424610726649249, −7.47420757709916163332268993789, −6.51049268056874839778112033554, −5.45901460375325237500302622062, −4.26070063880055380678030553923, −3.71301670186060922436333326105, −2.66223855023195415361548740522, −0.48694326412182266934276609103,
1.69017078653650306404017444731, 2.69233575885012420473105664725, 3.63760929399028963353355971717, 5.26064188261127967646138343871, 6.26310385656869515477935385730, 6.55634735269529140392997718882, 7.889406976582373192816151685303, 8.564949041518089805754635759787, 9.296321470081883441528010339347, 9.984664211648505456465607443119