L(s) = 1 | + (0.235 + 0.971i)2-s + (−0.327 − 0.945i)3-s + (−0.888 + 0.458i)4-s + (−0.262 + 0.105i)5-s + (0.841 − 0.540i)6-s + (−1.94 − 1.79i)7-s + (−0.654 − 0.755i)8-s + (−0.786 + 0.618i)9-s + (−0.164 − 0.230i)10-s + (0.182 − 0.751i)11-s + (0.723 + 0.690i)12-s + (−0.0379 + 0.0830i)13-s + (1.28 − 2.31i)14-s + (0.185 + 0.214i)15-s + (0.580 − 0.814i)16-s + (0.140 + 2.94i)17-s + ⋯ |
L(s) = 1 | + (0.166 + 0.687i)2-s + (−0.188 − 0.545i)3-s + (−0.444 + 0.229i)4-s + (−0.117 + 0.0470i)5-s + (0.343 − 0.220i)6-s + (−0.735 − 0.677i)7-s + (−0.231 − 0.267i)8-s + (−0.262 + 0.206i)9-s + (−0.0519 − 0.0729i)10-s + (0.0549 − 0.226i)11-s + (0.208 + 0.199i)12-s + (−0.0105 + 0.0230i)13-s + (0.343 − 0.618i)14-s + (0.0478 + 0.0552i)15-s + (0.145 − 0.203i)16-s + (0.0340 + 0.714i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.270502 + 0.668507i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.270502 + 0.668507i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.235 - 0.971i)T \) |
| 3 | \( 1 + (0.327 + 0.945i)T \) |
| 7 | \( 1 + (1.94 + 1.79i)T \) |
| 23 | \( 1 + (-1.20 - 4.64i)T \) |
good | 5 | \( 1 + (0.262 - 0.105i)T + (3.61 - 3.45i)T^{2} \) |
| 11 | \( 1 + (-0.182 + 0.751i)T + (-9.77 - 5.04i)T^{2} \) |
| 13 | \( 1 + (0.0379 - 0.0830i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.140 - 2.94i)T + (-16.9 + 1.61i)T^{2} \) |
| 19 | \( 1 + (0.369 - 7.76i)T + (-18.9 - 1.80i)T^{2} \) |
| 29 | \( 1 + (1.47 - 0.947i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.38 + 0.460i)T + (28.7 - 11.5i)T^{2} \) |
| 37 | \( 1 + (1.53 - 1.20i)T + (8.72 - 35.9i)T^{2} \) |
| 41 | \( 1 + (0.0592 - 0.412i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (2.78 - 3.21i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (3.32 + 5.75i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.993 + 0.0949i)T + (52.0 + 10.0i)T^{2} \) |
| 59 | \( 1 + (1.73 + 2.43i)T + (-19.2 + 55.7i)T^{2} \) |
| 61 | \( 1 + (3.39 - 9.81i)T + (-47.9 - 37.7i)T^{2} \) |
| 67 | \( 1 + (-8.56 + 8.16i)T + (3.18 - 66.9i)T^{2} \) |
| 71 | \( 1 + (-0.539 - 0.158i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (12.3 - 6.35i)T + (42.3 - 59.4i)T^{2} \) |
| 79 | \( 1 + (-12.1 + 1.15i)T + (77.5 - 14.9i)T^{2} \) |
| 83 | \( 1 + (-1.85 - 12.9i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (17.6 + 3.40i)T + (82.6 + 33.0i)T^{2} \) |
| 97 | \( 1 + (1.50 - 10.4i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.18649161151580195999452738362, −9.523633362102189446495101151334, −8.344287494260656792835847992557, −7.72655956459415696809200726509, −6.91954634610507146011027729807, −6.12632069087971552707154759233, −5.44065989001585857072071300205, −4.03015800387294298716786912361, −3.32750118031336480530073447168, −1.48966459386313645713621494414,
0.32833166492550307483320834942, 2.37737028355319045588659351830, 3.16464062859943334190671322042, 4.38095263274526185052131717883, 5.08330242982784620507698304921, 6.14382977128158947436503310899, 7.00481331829435620622615433435, 8.371215368118716779304558308536, 9.168448488847325074094450614096, 9.688846172805594407234538151747