| L(s) = 1 | + (1.40 − 0.123i)2-s + (0.151 − 0.0762i)3-s + (1.96 − 0.349i)4-s + (−1.59 − 3.60i)5-s + (0.204 − 0.126i)6-s + (−2.54 − 1.52i)7-s + (2.73 − 0.735i)8-s + (−1.76 + 2.38i)9-s + (−2.69 − 4.88i)10-s + (2.18 + 1.70i)11-s + (0.271 − 0.203i)12-s + (3.29 − 3.13i)13-s + (−3.77 − 1.83i)14-s + (−0.517 − 0.424i)15-s + (3.75 − 1.37i)16-s + (−2.87 − 3.50i)17-s + ⋯ |
| L(s) = 1 | + (0.996 − 0.0876i)2-s + (0.0874 − 0.0440i)3-s + (0.984 − 0.174i)4-s + (−0.714 − 1.61i)5-s + (0.0832 − 0.0515i)6-s + (−0.960 − 0.575i)7-s + (0.965 − 0.260i)8-s + (−0.589 + 0.795i)9-s + (−0.853 − 1.54i)10-s + (0.659 + 0.514i)11-s + (0.0784 − 0.0586i)12-s + (0.913 − 0.869i)13-s + (−1.00 − 0.489i)14-s + (−0.133 − 0.109i)15-s + (0.939 − 0.343i)16-s + (−0.698 − 0.851i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00816 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00816 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.54041 - 1.55304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54041 - 1.55304i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 + 0.123i)T \) |
| good | 3 | \( 1 + (-0.151 + 0.0762i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (1.59 + 3.60i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (2.54 + 1.52i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (-2.18 - 1.70i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-3.29 + 3.13i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (2.87 + 3.50i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.636 + 3.66i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (-4.70 - 2.22i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (3.24 + 1.83i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-1.67 + 0.332i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-2.48 - 1.57i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-6.38 - 7.04i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-2.25 - 6.83i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (-1.39 - 0.747i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (-1.03 + 0.586i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (-0.503 + 0.529i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.696 - 9.44i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (-9.94 - 8.57i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-14.4 - 2.14i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (9.10 - 5.45i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (15.1 + 4.60i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-9.33 + 5.91i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (-6.51 + 3.08i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (14.4 + 9.65i)T + (37.1 + 89.6i)T^{2} \) |
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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
−11.16044990346005529274333802725, −9.769827938739947460066769561940, −8.860023340977778730805049091638, −7.85896686760107784005120566241, −6.96588266649042746503768161618, −5.76502366535170320798878142505, −4.79924169523562973862123898220, −4.07842177123148140906861897081, −2.87829166236780123566131991507, −0.983444330842183605755945759136,
2.45513796610051776307170028622, 3.60443480375495132026603336412, 3.81209489751145007227391745794, 5.94438873744997373934852974988, 6.40915881679825822618996090956, 6.99092597821253562102410461407, 8.355972427695440071550285160485, 9.324003934202055579646428736124, 10.78036095620423355457706047590, 11.10290535681435635077065759561
