| L(s) = 1 | − 3.18i·2-s − 6.17·4-s − 7.66i·5-s − 10.1i·7-s + 6.93i·8-s − 24.4·10-s + 2.43·11-s + 18.1·13-s − 32.3·14-s − 2.56·16-s + 1.13·17-s − 12.3i·19-s + 47.3i·20-s − 7.76i·22-s + 24.2·23-s + ⋯ |
| L(s) = 1 | − 1.59i·2-s − 1.54·4-s − 1.53i·5-s − 1.44i·7-s + 0.867i·8-s − 2.44·10-s + 0.221·11-s + 1.39·13-s − 2.30·14-s − 0.160·16-s + 0.0668·17-s − 0.648i·19-s + 2.36i·20-s − 0.353i·22-s + 1.05·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.749160 + 1.38836i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.749160 + 1.38836i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (-23.6 - 35.9i)T \) |
| good | 2 | \( 1 + 3.18iT - 4T^{2} \) |
| 5 | \( 1 + 7.66iT - 25T^{2} \) |
| 7 | \( 1 + 10.1iT - 49T^{2} \) |
| 11 | \( 1 - 2.43T + 121T^{2} \) |
| 13 | \( 1 - 18.1T + 169T^{2} \) |
| 17 | \( 1 - 1.13T + 289T^{2} \) |
| 19 | \( 1 + 12.3iT - 361T^{2} \) |
| 23 | \( 1 - 24.2T + 529T^{2} \) |
| 29 | \( 1 - 35.5iT - 841T^{2} \) |
| 31 | \( 1 + 12.9T + 961T^{2} \) |
| 37 | \( 1 - 41.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 65.0T + 1.68e3T^{2} \) |
| 47 | \( 1 + 51.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 56.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 88.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 65.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 45.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 63.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 8.80iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 31.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 68.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 5.90iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 100.T + 9.40e3T^{2} \) |
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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
−10.89707466222839780943266188544, −9.653024327404394775697715696001, −9.031473401342137387227213201598, −8.121668162491066432300642015611, −6.70909179975941306948411513496, −5.02271645349805628952673603004, −4.24236149798732716194509152500, −3.34304546235729239688198208773, −1.37952035539078154410251679656, −0.812482486582303957766380338065,
2.47650315004521112998016422342, 3.84735649728922318701143926130, 5.55338294957120010873470978679, 6.07980389648772576467665271028, 6.84280296065780568494414872694, 7.83135366890773887469200725729, 8.705222571102166054134701828256, 9.516505875479894622647481210866, 10.87702899839125624941299413330, 11.52565662993983896283397778176
