| L(s) = 1 | + (1.72 − 2.05i)2-s + (0.734 + 2.01i)3-s + (−0.906 − 5.13i)4-s + (−1.71 + 1.43i)5-s + (5.41 + 1.97i)6-s + (−2.41 + 1.39i)7-s + (−7.48 − 4.32i)8-s + (−1.23 + 1.03i)9-s + (−0.00273 + 6.00i)10-s + (1.18 − 2.06i)11-s + (9.70 − 5.60i)12-s + (0.0138 − 0.0380i)13-s + (−1.29 + 7.36i)14-s + (−4.15 − 2.40i)15-s + (−12.0 + 4.37i)16-s + (−1.16 + 1.39i)17-s + ⋯ |
| L(s) = 1 | + (1.22 − 1.45i)2-s + (0.423 + 1.16i)3-s + (−0.453 − 2.56i)4-s + (−0.766 + 0.642i)5-s + (2.21 + 0.805i)6-s + (−0.910 + 0.525i)7-s + (−2.64 − 1.52i)8-s + (−0.410 + 0.344i)9-s + (−0.000865 + 1.89i)10-s + (0.358 − 0.621i)11-s + (2.80 − 1.61i)12-s + (0.00384 − 0.0105i)13-s + (−0.347 + 1.96i)14-s + (−1.07 − 0.620i)15-s + (−3.00 + 1.09i)16-s + (−0.283 + 0.337i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40987 - 0.788298i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40987 - 0.788298i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.71 - 1.43i)T \) |
| 19 | \( 1 + (-4.07 + 1.54i)T \) |
| good | 2 | \( 1 + (-1.72 + 2.05i)T + (-0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.734 - 2.01i)T + (-2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (2.41 - 1.39i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.18 + 2.06i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.0138 + 0.0380i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.16 - 1.39i)T + (-2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-2.50 + 0.441i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.25 - 1.89i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.44 - 2.49i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.227iT - 37T^{2} \) |
| 41 | \( 1 + (7.55 - 2.74i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-5.05 - 0.891i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (7.11 + 8.48i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (5.62 - 0.992i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (8.89 + 7.46i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.795 - 4.51i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (3.11 + 3.71i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.34 - 7.65i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.45 - 3.99i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-10.9 + 3.97i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.87 + 2.23i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-14.4 - 5.26i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.59 + 4.28i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−13.79946204051519124341030170717, −12.64991209412379971674836871395, −11.63735779885458112759076470018, −10.80660231608776319459610074401, −9.882889575405362721094484752099, −8.969616613696061351849084749265, −6.43216011982930921321896856973, −4.90543726888543826228538796154, −3.58690161690646589174210526780, −3.05863283512835045831298589419,
3.41905176049311243331850696719, 4.75141475409307554572941349715, 6.37732842081860600323055206655, 7.30488895826005835975013909772, 7.86841532253034256694493966791, 9.205552742063521069991936540986, 11.83638629492610364710018888467, 12.61202221998336038990368345742, 13.26559339090195010424056222604, 13.97956160031533784391038032375
