| L(s) = 1 | + (−0.599 − 1.28i)2-s + (0.147 + 0.551i)3-s + (−1.28 + 1.53i)4-s + (1.15 + 1.91i)5-s + (0.618 − 0.520i)6-s + (0.735 + 2.54i)7-s + (2.73 + 0.719i)8-s + (2.31 − 1.33i)9-s + (1.76 − 2.62i)10-s + (−3.42 − 1.98i)11-s + (−1.03 − 0.479i)12-s + (2.04 + 2.04i)13-s + (2.81 − 2.46i)14-s + (−0.888 + 0.918i)15-s + (−0.719 − 3.93i)16-s + (−0.155 − 0.580i)17-s + ⋯ |
| L(s) = 1 | + (−0.424 − 0.905i)2-s + (0.0853 + 0.318i)3-s + (−0.640 + 0.768i)4-s + (0.514 + 0.857i)5-s + (0.252 − 0.212i)6-s + (0.277 + 0.960i)7-s + (0.967 + 0.254i)8-s + (0.771 − 0.445i)9-s + (0.558 − 0.829i)10-s + (−1.03 − 0.597i)11-s + (−0.299 − 0.138i)12-s + (0.568 + 0.568i)13-s + (0.752 − 0.658i)14-s + (−0.229 + 0.237i)15-s + (−0.179 − 0.983i)16-s + (−0.0377 − 0.140i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.968397 + 0.0163474i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.968397 + 0.0163474i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.599 + 1.28i)T \) |
| 5 | \( 1 + (-1.15 - 1.91i)T \) |
| 7 | \( 1 + (-0.735 - 2.54i)T \) |
| good | 3 | \( 1 + (-0.147 - 0.551i)T + (-2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (3.42 + 1.98i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.04 - 2.04i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.155 + 0.580i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.00 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.39 - 0.640i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 7.25iT - 29T^{2} \) |
| 31 | \( 1 + (-2.83 - 1.63i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (9.29 + 2.49i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 3.35T + 41T^{2} \) |
| 43 | \( 1 + (3.31 - 3.31i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.24 + 12.1i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.90 + 0.779i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-7.41 + 12.8i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.11 + 3.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.67 - 0.984i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 4.86iT - 71T^{2} \) |
| 73 | \( 1 + (-9.44 + 2.53i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.45 - 5.98i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.46 - 5.46i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.35 - 1.93i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.11 + 1.11i)T - 97iT^{2} \) |
| 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
−13.12689671439597835864431586439, −11.92020321679702874369522220796, −11.01495139186743441057315114033, −10.16721418974019288768586672594, −9.241219375763108350920108848684, −8.244616616879961622500618264252, −6.75364512710845152193832980112, −5.17567608770292908630806188847, −3.50038730831371471685755954202, −2.19986676452856643319575936019,
1.40957235883127094120216417481, 4.46553662604100648614585640000, 5.40973049162239723873879057885, 6.91117484020116057726213277922, 7.82001654463975959017747291429, 8.695823386111270160116814592103, 10.14528994425466790490075247607, 10.53998345301821139550807114992, 12.65061025530135155266625066523, 13.28998336205367192772561837611
