| L(s) = 1 | + (1.01 + 0.982i)2-s + (−1.25 − 1.88i)3-s + (0.0682 + 1.99i)4-s + (1.55 + 0.309i)5-s + (0.570 − 3.14i)6-s + (3.91 − 0.779i)7-s + (−1.89 + 2.09i)8-s + (−0.812 + 1.96i)9-s + (1.27 + 1.84i)10-s + (−2.04 − 1.36i)11-s + (3.67 − 2.64i)12-s + (−1.67 + 1.67i)13-s + (4.74 + 3.05i)14-s + (−1.37 − 3.32i)15-s + (−3.99 + 0.272i)16-s + (−0.0531 + 4.12i)17-s + ⋯ |
| L(s) = 1 | + (0.719 + 0.694i)2-s + (−0.726 − 1.08i)3-s + (0.0341 + 0.999i)4-s + (0.696 + 0.138i)5-s + (0.233 − 1.28i)6-s + (1.48 − 0.294i)7-s + (−0.670 + 0.742i)8-s + (−0.270 + 0.654i)9-s + (0.404 + 0.583i)10-s + (−0.615 − 0.411i)11-s + (1.06 − 0.762i)12-s + (−0.464 + 0.464i)13-s + (1.26 + 0.817i)14-s + (−0.355 − 0.857i)15-s + (−0.997 + 0.0681i)16-s + (−0.0128 + 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.41711 + 0.189922i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41711 + 0.189922i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.01 - 0.982i)T \) |
| 17 | \( 1 + (0.0531 - 4.12i)T \) |
| good | 3 | \( 1 + (1.25 + 1.88i)T + (-1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (-1.55 - 0.309i)T + (4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (-3.91 + 0.779i)T + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (2.04 + 1.36i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (1.67 - 1.67i)T - 13iT^{2} \) |
| 19 | \( 1 + (1.36 + 3.28i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (5.72 + 3.82i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (0.362 - 1.82i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (-5.41 - 8.09i)T + (-11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (-5.16 + 3.44i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (0.977 + 4.91i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (1.08 - 2.61i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (2.87 - 2.87i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.16 + 1.31i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-6.57 - 2.72i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-1.72 - 8.69i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + 6.11iT - 67T^{2} \) |
| 71 | \( 1 + (5.58 - 3.73i)T + (27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-2.78 + 13.9i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-7.05 + 10.5i)T + (-30.2 - 72.9i)T^{2} \) |
| 83 | \( 1 + (16.0 - 6.62i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-0.443 + 0.443i)T - 89iT^{2} \) |
| 97 | \( 1 + (-9.17 - 1.82i)T + (89.6 + 37.1i)T^{2} \) |
| 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
−13.37136257482245713416502480082, −12.39177188706281956013017502606, −11.58059008864669431511768068313, −10.54838160082916980529216922500, −8.529030816480730553398339474516, −7.65198398978686303052310881599, −6.57528634024724014752236510281, −5.68211508438430296521084849409, −4.52844978291292769826138381259, −2.07893119897008648337159339561,
2.16392410407378483676235178014, 4.28291324406234693293556453209, 5.16488778503566802514940006512, 5.80699022984473531930389463054, 7.938229147819725286426033445730, 9.792718250671176428287476808968, 10.04731657408218742567291345340, 11.37504612088177485583472656450, 11.75412242875377167912724220254, 13.17032124434650177723340800178
