| L(s) = 1 | + (1.25 − 0.659i)2-s + (0.977 − 1.42i)3-s + (1.13 − 1.64i)4-s + (−0.765 − 2.25i)5-s + (0.280 − 2.43i)6-s + (1.05 + 1.37i)7-s + (0.326 − 2.80i)8-s + (−1.08 − 2.79i)9-s + (−2.44 − 2.31i)10-s + (−0.555 + 0.0364i)11-s + (−1.25 − 3.22i)12-s + (4.12 + 4.70i)13-s + (2.22 + 1.02i)14-s + (−3.97 − 1.11i)15-s + (−1.44 − 3.73i)16-s + (−4.04 + 4.04i)17-s + ⋯ |
| L(s) = 1 | + (0.884 − 0.466i)2-s + (0.564 − 0.825i)3-s + (0.565 − 0.824i)4-s + (−0.342 − 1.00i)5-s + (0.114 − 0.993i)6-s + (0.398 + 0.519i)7-s + (0.115 − 0.993i)8-s + (−0.362 − 0.932i)9-s + (−0.773 − 0.732i)10-s + (−0.167 + 0.0109i)11-s + (−0.361 − 0.932i)12-s + (1.14 + 1.30i)13-s + (0.594 + 0.273i)14-s + (−1.02 − 0.286i)15-s + (−0.361 − 0.932i)16-s + (−0.981 + 0.981i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48798 - 2.39854i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48798 - 2.39854i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.25 + 0.659i)T \) |
| 3 | \( 1 + (-0.977 + 1.42i)T \) |
| good | 5 | \( 1 + (0.765 + 2.25i)T + (-3.96 + 3.04i)T^{2} \) |
| 7 | \( 1 + (-1.05 - 1.37i)T + (-1.81 + 6.76i)T^{2} \) |
| 11 | \( 1 + (0.555 - 0.0364i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (-4.12 - 4.70i)T + (-1.69 + 12.8i)T^{2} \) |
| 17 | \( 1 + (4.04 - 4.04i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.619 - 3.11i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (2.14 + 1.64i)T + (5.95 + 22.2i)T^{2} \) |
| 29 | \( 1 + (-0.508 - 1.03i)T + (-17.6 + 23.0i)T^{2} \) |
| 31 | \( 1 + (-2.53 + 1.46i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.55 + 7.80i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-6.43 - 4.93i)T + (10.6 + 39.6i)T^{2} \) |
| 43 | \( 1 + (-0.753 - 11.4i)T + (-42.6 + 5.61i)T^{2} \) |
| 47 | \( 1 + (1.75 + 6.53i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (2.94 + 4.40i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-2.78 - 8.19i)T + (-46.8 + 35.9i)T^{2} \) |
| 61 | \( 1 + (6.59 - 3.25i)T + (37.1 - 48.3i)T^{2} \) |
| 67 | \( 1 + (0.254 - 3.88i)T + (-66.4 - 8.74i)T^{2} \) |
| 71 | \( 1 + (1.77 - 4.29i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (5.93 + 14.3i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-9.13 + 2.44i)T + (68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (-13.0 - 4.43i)T + (65.8 + 50.5i)T^{2} \) |
| 89 | \( 1 + (1.86 + 0.773i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (-11.7 - 6.77i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.78085139558619603150829855063, −9.347474403745806979374053245673, −8.678972490224089869376422607741, −7.890441521656458072430538236251, −6.52722672930832285665320873734, −5.92143094999002045801135407459, −4.53906323575047047840123983926, −3.80751538898531051104796536949, −2.25152340712468999472862743143, −1.32935270112232484882731743908,
2.64795333581700559927628767228, 3.37653534196922407810798349968, 4.34507691625736886879050908565, 5.29404070682882961076555917274, 6.46414478892965405852908373601, 7.48362339126222089165684774638, 8.090288684968021617750618195205, 9.117228458319951791741802608235, 10.54857708500266771905111007778, 10.90279701913262168212556233326
