| L(s) = 1 | − 2.49i·2-s − 4.20·4-s + 1.23·5-s + 5.49i·8-s − 3.07i·10-s − 5.49i·11-s + 2.96i·13-s + 5.26·16-s + 4.31·17-s − 5.55i·19-s − 5.19·20-s − 13.6·22-s − 1.63i·23-s − 3.47·25-s + 7.39·26-s + ⋯ |
| L(s) = 1 | − 1.76i·2-s − 2.10·4-s + 0.552·5-s + 1.94i·8-s − 0.973i·10-s − 1.65i·11-s + 0.823i·13-s + 1.31·16-s + 1.04·17-s − 1.27i·19-s − 1.16·20-s − 2.91·22-s − 0.340i·23-s − 0.694·25-s + 1.44·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.251309701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.251309701\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.49iT - 2T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 + 5.49iT - 11T^{2} \) |
| 13 | \( 1 - 2.96iT - 13T^{2} \) |
| 17 | \( 1 - 4.31T + 17T^{2} \) |
| 19 | \( 1 + 5.55iT - 19T^{2} \) |
| 23 | \( 1 + 1.63iT - 23T^{2} \) |
| 29 | \( 1 - 0.509iT - 29T^{2} \) |
| 31 | \( 1 + 8.13iT - 31T^{2} \) |
| 37 | \( 1 + 3.06T + 37T^{2} \) |
| 41 | \( 1 - 0.354T + 41T^{2} \) |
| 43 | \( 1 + 0.0637T + 43T^{2} \) |
| 47 | \( 1 + 9.86T + 47T^{2} \) |
| 53 | \( 1 - 4.12iT - 53T^{2} \) |
| 59 | \( 1 + 3.07T + 59T^{2} \) |
| 61 | \( 1 + 6.89iT - 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 4.63iT - 71T^{2} \) |
| 73 | \( 1 - 7.03iT - 73T^{2} \) |
| 79 | \( 1 + 0.331T + 79T^{2} \) |
| 83 | \( 1 - 7.39T + 83T^{2} \) |
| 89 | \( 1 - 3.43T + 89T^{2} \) |
| 97 | \( 1 + 4.45iT - 97T^{2} \) |
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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
−9.344008725759967721983680275922, −8.790451021647846506459234463308, −7.84689277426523328407804531844, −6.43834258396300005887509345743, −5.56296677254395305409152539291, −4.56302680884048015428039140263, −3.55699441621634169609844144651, −2.80047228523413379968931986926, −1.74277712529403555390153935360, −0.53060830915269153037286455638,
1.68291458774730828601538410132, 3.44565926102029812851661673199, 4.62299696898213982543838839841, 5.37034331416202039262188533080, 5.99574445727503336207775763364, 6.90999513693391774097917469741, 7.63244836143374888075815393726, 8.147333263458452857540384715839, 9.180365662399917719909484193036, 9.956932290735625101572398578964
