| L(s) = 1 | + 1.17·5-s − i·7-s + 2.90i·11-s + 1.05i·13-s − 6.48i·17-s + 3.11·19-s + 0.812·23-s − 3.61·25-s − 4.03·29-s − 0.108i·31-s − 1.17i·35-s − 9.97i·37-s − 9.44i·41-s − 10.3·43-s + 4.03·47-s + ⋯ |
| L(s) = 1 | + 0.526·5-s − 0.377i·7-s + 0.876i·11-s + 0.292i·13-s − 1.57i·17-s + 0.714·19-s + 0.169·23-s − 0.722·25-s − 0.749·29-s − 0.0195i·31-s − 0.199i·35-s − 1.63i·37-s − 1.47i·41-s − 1.57·43-s + 0.588·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0828 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0828 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.559017072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.559017072\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.17T + 5T^{2} \) |
| 11 | \( 1 - 2.90iT - 11T^{2} \) |
| 13 | \( 1 - 1.05iT - 13T^{2} \) |
| 17 | \( 1 + 6.48iT - 17T^{2} \) |
| 19 | \( 1 - 3.11T + 19T^{2} \) |
| 23 | \( 1 - 0.812T + 23T^{2} \) |
| 29 | \( 1 + 4.03T + 29T^{2} \) |
| 31 | \( 1 + 0.108iT - 31T^{2} \) |
| 37 | \( 1 + 9.97iT - 37T^{2} \) |
| 41 | \( 1 + 9.44iT - 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 4.03T + 47T^{2} \) |
| 53 | \( 1 + 2.97T + 53T^{2} \) |
| 59 | \( 1 - 0.868iT - 59T^{2} \) |
| 61 | \( 1 - 5.33iT - 61T^{2} \) |
| 67 | \( 1 - 3.97T + 67T^{2} \) |
| 71 | \( 1 + 0.844T + 71T^{2} \) |
| 73 | \( 1 - 3.59T + 73T^{2} \) |
| 79 | \( 1 + 9.48iT - 79T^{2} \) |
| 83 | \( 1 + 2.71iT - 83T^{2} \) |
| 89 | \( 1 - 2.22iT - 89T^{2} \) |
| 97 | \( 1 + 3.93T + 97T^{2} \) |
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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
−7.49381128562352746193424250204, −7.39011187377609089263703003887, −6.58482465700245207477057926478, −5.60466634370029367791265735793, −5.11270502341315484627204943046, −4.25782226963919375495757773730, −3.44732591179153770361503258422, −2.41451422544836830823500424195, −1.69578577194564960015886786696, −0.39436000208568442140695743500,
1.18687152758201395290321768583, 2.02456135982511011405874266653, 3.11628213676369231682291807573, 3.66641087881266336421510158088, 4.77835022820534296798679423391, 5.51634365413219132374463290692, 6.12236796648976462545975784810, 6.61562329957144242322608847034, 7.75518548019607080265908968862, 8.254906224892074369703595623606
