L(s) = 1 | − i·3-s + 1.73i·5-s − 9-s − i·11-s + 1.73i·13-s + 1.73·15-s + 17-s + i·19-s − 1.73·23-s − 1.99·25-s + i·27-s − 33-s + 1.73·39-s − 41-s + i·43-s + ⋯ |
L(s) = 1 | − i·3-s + 1.73i·5-s − 9-s − i·11-s + 1.73i·13-s + 1.73·15-s + 17-s + i·19-s − 1.73·23-s − 1.99·25-s + i·27-s − 33-s + 1.73·39-s − 41-s + i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9810866803\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9810866803\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 1.73iT - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 13 | \( 1 - 1.73iT - T^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 + 1.73T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - 2iT - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.804390179707340998928802375423, −8.007759280248985988989656898860, −7.44986974490388435525083439309, −6.68493743347155157805162107260, −6.17986913264863319532358670495, −5.63771301072083091980415140643, −4.00636729930329663761055668099, −3.34478495687601401808718040363, −2.43561375941112266626914909062, −1.59817132773465775909791077745,
0.57846866493989287438974716546, 2.03672643876248542983809338444, 3.29960000589114888021866666765, 4.15119737514811157741690950143, 4.92202460233073808487471143664, 5.35722869105183291711686357844, 6.03922245919744545394966810682, 7.50518335530158334105500466828, 8.116799338303557392384019554625, 8.701211754624978364771878490885