L(s) = 1 | + (−0.965 − 0.258i)3-s + (2.25 − 1.38i)7-s + (0.866 + 0.499i)9-s + (−2.42 − 4.19i)11-s + (2.27 − 2.27i)13-s + (0.574 − 2.14i)17-s + (−0.107 + 0.185i)19-s + (−2.53 + 0.758i)21-s + (−6.64 + 1.78i)23-s + (−0.707 − 0.707i)27-s − 9.28i·29-s + (−7.78 + 4.49i)31-s + (1.25 + 4.68i)33-s + (2.62 + 9.78i)37-s + (−2.78 + 1.61i)39-s + ⋯ |
L(s) = 1 | + (−0.557 − 0.149i)3-s + (0.851 − 0.524i)7-s + (0.288 + 0.166i)9-s + (−0.730 − 1.26i)11-s + (0.631 − 0.631i)13-s + (0.139 − 0.520i)17-s + (−0.0246 + 0.0426i)19-s + (−0.553 + 0.165i)21-s + (−1.38 + 0.371i)23-s + (−0.136 − 0.136i)27-s − 1.72i·29-s + (−1.39 + 0.807i)31-s + (0.218 + 0.815i)33-s + (0.431 + 1.60i)37-s + (−0.446 + 0.257i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9874716678\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9874716678\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.25 + 1.38i)T \) |
good | 11 | \( 1 + (2.42 + 4.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.27 + 2.27i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.574 + 2.14i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.107 - 0.185i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.64 - 1.78i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 9.28iT - 29T^{2} \) |
| 31 | \( 1 + (7.78 - 4.49i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.62 - 9.78i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 2.32iT - 41T^{2} \) |
| 43 | \( 1 + (-5.01 - 5.01i)T + 43iT^{2} \) |
| 47 | \( 1 + (-9.36 + 2.50i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.785 + 2.93i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (2.68 + 4.64i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (13.2 + 7.63i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.97 + 1.33i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + (0.146 + 0.0391i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.0599 + 0.0346i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.467 - 0.467i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.997 - 1.72i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.51 + 7.51i)T + 97iT^{2} \) |
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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
−8.581983596209753388468302706583, −7.934122525029438068775935370840, −7.47733507752426237389995541805, −6.18384040650702067755090688773, −5.74518544351834153624690363599, −4.86306137359385776869364505802, −3.92852473248881301980099273615, −2.90276231392705139654554223373, −1.53326537047484831587043093521, −0.37613351865726204497875627918,
1.58686306067769017848887468579, 2.36160008560493299411867973359, 3.96378715502650547878105812172, 4.52241651483471894462914784476, 5.56154391309607219191512010277, 5.97686208425366699684537031924, 7.30115559475728672869048528176, 7.60997979128986131589090514784, 8.828684528051178775438629728923, 9.221833254534838126153503618786