| L(s) = 1 | + i·2-s − 4-s − 0.266i·5-s + (1.34 − 2.27i)7-s − i·8-s + 0.266·10-s + (−1.62 + 2.89i)11-s − 2.55·13-s + (2.27 + 1.34i)14-s + 16-s − 4.96·17-s + 8.21·19-s + 0.266i·20-s + (−2.89 − 1.62i)22-s + 5.23·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.118i·5-s + (0.509 − 0.860i)7-s − 0.353i·8-s + 0.0841·10-s + (−0.489 + 0.871i)11-s − 0.707·13-s + (0.608 + 0.360i)14-s + 0.250·16-s − 1.20·17-s + 1.88·19-s + 0.0594i·20-s + (−0.616 − 0.346i)22-s + 1.09·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.600669998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.600669998\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.34 + 2.27i)T \) |
| 11 | \( 1 + (1.62 - 2.89i)T \) |
| good | 5 | \( 1 + 0.266iT - 5T^{2} \) |
| 13 | \( 1 + 2.55T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 19 | \( 1 - 8.21T + 19T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 - 5.25iT - 29T^{2} \) |
| 31 | \( 1 + 4.28iT - 31T^{2} \) |
| 37 | \( 1 - 7.39T + 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 43 | \( 1 + 1.46iT - 43T^{2} \) |
| 47 | \( 1 + 10.0iT - 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 9.10iT - 59T^{2} \) |
| 61 | \( 1 + 5.98T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 4.93T + 71T^{2} \) |
| 73 | \( 1 + 2.96T + 73T^{2} \) |
| 79 | \( 1 - 6.16iT - 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 + 1.78iT - 89T^{2} \) |
| 97 | \( 1 - 0.146iT - 97T^{2} \) |
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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
−9.517556808239507887790535844947, −8.831917589711080600318616303826, −7.71074210169718727822852489060, −7.30413746484010687924427401349, −6.63914371375435702296535555226, −5.15591486682274830743872908637, −4.91639143114671358849966011850, −3.83349215939229233556685191381, −2.47680551622483685105695275867, −0.897088355712510528349604892895,
0.995423717896900190587592871950, 2.53759644644839318696201242202, 2.99913684759682869242078466188, 4.46506202179447377867142045593, 5.19895734467539489931300546408, 5.98633987832385320361056200271, 7.22328149070144801794855497678, 8.028642119705733093460574024653, 8.960011144380557342254521613729, 9.366600008764101782234305886888
