L(s) = 1 | + 7-s − i·11-s + i·13-s − 19-s + 23-s + i·29-s − i·31-s + 37-s + i·41-s + i·43-s − i·47-s + 49-s + i·53-s + 59-s − i·61-s + ⋯ |
L(s) = 1 | + 7-s − i·11-s + i·13-s − 19-s + 23-s + i·29-s − i·31-s + 37-s + i·41-s + i·43-s − i·47-s + 49-s + i·53-s + 59-s − i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.937 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.937 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.401838157 + 0.4300093812i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.401838157 + 0.4300093812i\) |
\(L(1)\) |
\(\approx\) |
\(1.243596501 + 0.04050903117i\) |
\(L(1)\) |
\(\approx\) |
\(1.243596501 + 0.04050903117i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
good | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 \) |
| 67 | \( 1 \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.09109894375827589117523967546, −20.783277231156725187702965799356, −19.85914943287757605737691818950, −19.06608110335960054394684578811, −18.03839371522105825448338697793, −17.52916216152038982343425575964, −16.902721255559957725067920224780, −15.64557597769380464501194937545, −15.028960144230533249192505468652, −14.46466518846911655931945391904, −13.36246982873655537823210421108, −12.6247959749836261900337178267, −11.834920626388016934407679110429, −10.84774091250862767874552462023, −10.29591237462059365426422334648, −9.20495370775068627971595520926, −8.317109995251592920600040933653, −7.59733871930447588191903206037, −6.729863449477612269051058842580, −5.56517894365207004789925743020, −4.81034894992068596502858645351, −3.98621561400878084182195245421, −2.68258116519757576041417560291, −1.81095157201948800682794871000, −0.656854326214344873609933291305,
0.83851140030608451538261242778, 1.83576628091744590449221964619, 2.89156919308596179534264676658, 4.08748117388278184447876328721, 4.80839971011998117591018280288, 5.83662877624228762097370175283, 6.70153433448390430936858129219, 7.704042825067255333165681564349, 8.56287705753642100281012664180, 9.148546995310465729707632408601, 10.35328491482922695657133982934, 11.29097061892776858526251090587, 11.541186854277688015805688587097, 12.83691529209321990398348802395, 13.53258572234594174434000489686, 14.55045755274230389257999215170, 14.857231737759181842312870644843, 16.119037769136766413168687912580, 16.76964009973404531811172882026, 17.45546099992024669738518608203, 18.524883247231907052474183580290, 18.93103491206463671928568327340, 19.92098521393753846380185006135, 20.799701087381542573936695341866, 21.564595924207365870329047295380