L(s) = 1 | + (−1.56 + 0.747i)3-s + (−2.60 − 0.456i)7-s + (1.88 − 2.33i)9-s + (−0.698 + 0.403i)11-s + 3.86·13-s + (−1.83 + 1.05i)17-s + (−2.70 − 1.56i)19-s + (4.41 − 1.23i)21-s + (−2.43 + 4.21i)23-s + (−1.19 + 5.05i)27-s − 6.67i·29-s + (5.65 − 3.26i)31-s + (0.789 − 1.15i)33-s + (−6.75 − 3.89i)37-s + (−6.04 + 2.89i)39-s + ⋯ |
L(s) = 1 | + (−0.902 + 0.431i)3-s + (−0.985 − 0.172i)7-s + (0.627 − 0.778i)9-s + (−0.210 + 0.121i)11-s + 1.07·13-s + (−0.445 + 0.257i)17-s + (−0.620 − 0.358i)19-s + (0.962 − 0.269i)21-s + (−0.508 + 0.879i)23-s + (−0.229 + 0.973i)27-s − 1.23i·29-s + (1.01 − 0.585i)31-s + (0.137 − 0.200i)33-s + (−1.11 − 0.641i)37-s + (−0.967 + 0.462i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8470281299\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8470281299\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.56 - 0.747i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.60 + 0.456i)T \) |
good | 11 | \( 1 + (0.698 - 0.403i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.86T + 13T^{2} \) |
| 17 | \( 1 + (1.83 - 1.05i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.70 + 1.56i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.43 - 4.21i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6.67iT - 29T^{2} \) |
| 31 | \( 1 + (-5.65 + 3.26i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.75 + 3.89i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.44T + 41T^{2} \) |
| 43 | \( 1 - 0.819iT - 43T^{2} \) |
| 47 | \( 1 + (2.40 + 1.38i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.67 - 11.5i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.86 - 4.95i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.79 - 1.03i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.44 + 5.45i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (1.54 + 2.67i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.76 - 11.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.8iT - 83T^{2} \) |
| 89 | \( 1 + (1.60 - 2.78i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 1.01T + 97T^{2} \) |
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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
−9.410992823161356000887195852563, −8.639915135507512070577232891517, −7.58796921106638841147499787520, −6.65521521603833356007020860928, −6.11446198114162898472231020581, −5.46007876274726447153600466060, −4.18221146626506825279477560338, −3.82553345925299379470908293666, −2.47260731207315555683496079580, −0.877093613061825733639996080922,
0.45645901531443444875326514653, 1.82664390308841483137284488680, 3.04821480587795671947587962433, 4.09758210862497706545340002102, 5.07348376243670682189460593315, 5.97735789508097205877532980293, 6.52439122995585036244076781107, 7.09185902373976273759936895118, 8.287417204796216643700474286817, 8.784777216043667387399937619288