| L(s) = 1 | + (−1.51 + 0.847i)3-s + (−1.16 − 1.90i)5-s + (1.56 − 2.55i)9-s − 3.03i·11-s − 2.31·13-s + (3.38 + 1.89i)15-s + 5.19i·17-s + 6.44i·19-s − 4.86·23-s + (−2.27 + 4.45i)25-s + (−0.194 + 5.19i)27-s − 3.48i·29-s + 1.34i·31-s + (2.57 + 4.58i)33-s − 2.81i·37-s + ⋯ |
| L(s) = 1 | + (−0.872 + 0.489i)3-s + (−0.522 − 0.852i)5-s + (0.521 − 0.853i)9-s − 0.915i·11-s − 0.642·13-s + (0.872 + 0.488i)15-s + 1.25i·17-s + 1.47i·19-s − 1.01·23-s + (−0.454 + 0.890i)25-s + (−0.0374 + 0.999i)27-s − 0.647i·29-s + 0.241i·31-s + (0.447 + 0.798i)33-s − 0.462i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8753074066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8753074066\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.51 - 0.847i)T \) |
| 5 | \( 1 + (1.16 + 1.90i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3.03iT - 11T^{2} \) |
| 13 | \( 1 + 2.31T + 13T^{2} \) |
| 17 | \( 1 - 5.19iT - 17T^{2} \) |
| 19 | \( 1 - 6.44iT - 19T^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 + 3.48iT - 29T^{2} \) |
| 31 | \( 1 - 1.34iT - 31T^{2} \) |
| 37 | \( 1 + 2.81iT - 37T^{2} \) |
| 41 | \( 1 - 1.06T + 41T^{2} \) |
| 43 | \( 1 - 2.42iT - 43T^{2} \) |
| 47 | \( 1 + 3.81iT - 47T^{2} \) |
| 53 | \( 1 - 7.05T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 13.2iT - 61T^{2} \) |
| 67 | \( 1 - 4.29iT - 67T^{2} \) |
| 71 | \( 1 + 7.08iT - 71T^{2} \) |
| 73 | \( 1 + 0.152T + 73T^{2} \) |
| 79 | \( 1 - 6.08T + 79T^{2} \) |
| 83 | \( 1 - 10.2iT - 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 8.63T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.615435170280891930558014834391, −8.198595076036011930708996530473, −7.32231061644767537128541080548, −6.15865532879806434785397337385, −5.76230511584819091461757499139, −4.94349753958592905819998237687, −3.97694058519442529819681713233, −3.62501622935065455438463713038, −1.86801180247913575880598475457, −0.64173019581152845379561336617,
0.53859360078314714631931159979, 2.13223605696486187802792174261, 2.84344181095685933266090803680, 4.22072934206320867852641455046, 4.83954991909039158637063254817, 5.69440009824739166539696657937, 6.72969738899195399653106549762, 7.16802189500961626516852435005, 7.55670974395928508853655128085, 8.634321158755926658057718937542
