L(s) = 1 | + i·2-s + (−1.61 − 0.618i)3-s − 4-s − 3.23·5-s + (0.618 − 1.61i)6-s + (0.381 − 2.61i)7-s − i·8-s + (2.23 + 2.00i)9-s − 3.23i·10-s − i·11-s + (1.61 + 0.618i)12-s + 6i·13-s + (2.61 + 0.381i)14-s + (5.23 + 2.00i)15-s + 16-s + 7.23·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.934 − 0.356i)3-s − 0.5·4-s − 1.44·5-s + (0.252 − 0.660i)6-s + (0.144 − 0.989i)7-s − 0.353i·8-s + (0.745 + 0.666i)9-s − 1.02i·10-s − 0.301i·11-s + (0.467 + 0.178i)12-s + 1.66i·13-s + (0.699 + 0.102i)14-s + (1.35 + 0.516i)15-s + 0.250·16-s + 1.75·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.607834 + 0.356572i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.607834 + 0.356572i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (1.61 + 0.618i)T \) |
| 7 | \( 1 + (-0.381 + 2.61i)T \) |
| 11 | \( 1 + iT \) |
good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 7.23T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 1.23iT - 23T^{2} \) |
| 29 | \( 1 - 0.472iT - 29T^{2} \) |
| 31 | \( 1 - 0.763iT - 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 - 7.70iT - 53T^{2} \) |
| 59 | \( 1 - 3.23T + 59T^{2} \) |
| 61 | \( 1 + 3.52iT - 61T^{2} \) |
| 67 | \( 1 - 1.52T + 67T^{2} \) |
| 71 | \( 1 + 6.18iT - 71T^{2} \) |
| 73 | \( 1 - 14.1iT - 73T^{2} \) |
| 79 | \( 1 - 7.23T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 2.47iT - 97T^{2} \) |
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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
−11.29962584822585828644673051019, −10.51113343226774783540031280369, −9.350064208960959844459114412993, −7.960195100297008361430514007303, −7.50927110551669740277263008710, −6.77201429867421611585224717694, −5.63996305178035765135593991040, −4.41055849278307176460139108080, −3.80645174564738535204300583265, −1.01607175494390905174026577253,
0.69097073518968889309072920179, 2.96460290260980045292561963437, 3.97153044460235755914528851188, 5.11231936387832528042773213988, 5.82434976170426261929661264938, 7.45913893790855839235613836933, 8.141890011241811922362863355993, 9.311694145412143987230738137314, 10.28957020180055898715842435896, 11.02084451659308592347498418975