| L(s) = 1 | + (0.393 + 0.493i)2-s + (−0.988 − 0.149i)3-s + (0.356 − 1.56i)4-s + (1.89 + 1.29i)5-s + (−0.315 − 0.547i)6-s + (0.505 − 0.875i)7-s + (2.04 − 0.987i)8-s + (0.955 + 0.294i)9-s + (0.108 + 1.44i)10-s + (0.296 + 1.30i)11-s + (−0.584 + 1.49i)12-s + (−0.0191 + 0.255i)13-s + (0.631 − 0.0951i)14-s + (−1.67 − 1.55i)15-s + (−1.58 − 0.765i)16-s + (1.44 − 0.987i)17-s + ⋯ |
| L(s) = 1 | + (0.278 + 0.349i)2-s + (−0.570 − 0.0860i)3-s + (0.178 − 0.780i)4-s + (0.846 + 0.577i)5-s + (−0.128 − 0.223i)6-s + (0.190 − 0.330i)7-s + (0.724 − 0.348i)8-s + (0.318 + 0.0982i)9-s + (0.0342 + 0.456i)10-s + (0.0895 + 0.392i)11-s + (−0.168 + 0.430i)12-s + (−0.00530 + 0.0707i)13-s + (0.168 − 0.0254i)14-s + (−0.433 − 0.402i)15-s + (−0.397 − 0.191i)16-s + (0.351 − 0.239i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22979 + 0.0139326i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22979 + 0.0139326i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.988 + 0.149i)T \) |
| 43 | \( 1 + (-0.379 + 6.54i)T \) |
| good | 2 | \( 1 + (-0.393 - 0.493i)T + (-0.445 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-1.89 - 1.29i)T + (1.82 + 4.65i)T^{2} \) |
| 7 | \( 1 + (-0.505 + 0.875i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.296 - 1.30i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (0.0191 - 0.255i)T + (-12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (-1.44 + 0.987i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (7.50 - 2.31i)T + (15.6 - 10.7i)T^{2} \) |
| 23 | \( 1 + (3.00 - 2.78i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (3.25 - 0.490i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (0.345 - 0.881i)T + (-22.7 - 21.0i)T^{2} \) |
| 37 | \( 1 + (1.91 + 3.31i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.72 - 9.68i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (0.819 - 3.58i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (-0.209 - 2.79i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (11.0 + 5.31i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-3.50 - 8.94i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-4.09 + 1.26i)T + (55.3 - 37.7i)T^{2} \) |
| 71 | \( 1 + (-1.72 - 1.60i)T + (5.30 + 70.8i)T^{2} \) |
| 73 | \( 1 + (-0.940 + 12.5i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-6.96 + 12.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.79 - 1.32i)T + (79.3 + 24.4i)T^{2} \) |
| 89 | \( 1 + (9.67 + 1.45i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + (0.381 + 1.67i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−13.55863102031445680178615601060, −12.41149202051009317581307970224, −11.00068322067187637036997170996, −10.38825891008994779261335720651, −9.474749011457134719051847969735, −7.57644434625000697940252670741, −6.44003547557895401744229369204, −5.77142359974244749260149175781, −4.39750488505352128797680586103, −1.90295025202712584287440340818,
2.16248270584587633738553051933, 4.07489813341531856444603765417, 5.36691880911863899687706407210, 6.54717747424782514715318917581, 8.109095485769875931735155536799, 9.106892503681223903649786225992, 10.48049712881744431515804855835, 11.37315512135454355966001875408, 12.47407624263265776929240384185, 12.98814122217914707582825049472
