L(s) = 1 | + (−1.18 − 1.26i)3-s + (0.686 − 1.18i)5-s + (0.5 + 0.866i)7-s + (−0.186 + 2.99i)9-s + (2.18 + 3.78i)11-s + (−1 + 1.73i)13-s + (−2.31 + 0.543i)15-s − 4.37·17-s − 5·19-s + (0.5 − 1.65i)21-s + (−3.68 + 6.38i)23-s + (1.55 + 2.69i)25-s + (4.00 − 3.31i)27-s + (−1.37 − 2.37i)29-s + (1 − 1.73i)31-s + ⋯ |
L(s) = 1 | + (−0.684 − 0.728i)3-s + (0.306 − 0.531i)5-s + (0.188 + 0.327i)7-s + (−0.0620 + 0.998i)9-s + (0.659 + 1.14i)11-s + (−0.277 + 0.480i)13-s + (−0.597 + 0.140i)15-s − 1.06·17-s − 1.14·19-s + (0.109 − 0.361i)21-s + (−0.768 + 1.33i)23-s + (0.311 + 0.539i)25-s + (0.769 − 0.638i)27-s + (−0.254 − 0.441i)29-s + (0.179 − 0.311i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9074835849\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9074835849\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.18 + 1.26i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.686 + 1.18i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.18 - 3.78i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + (3.68 - 6.38i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.37 + 2.37i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (5.18 - 8.98i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.55 - 7.89i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 2.74T + 53T^{2} \) |
| 59 | \( 1 + (-3.55 + 6.16i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.05 - 12.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.55 + 13.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 5.11T + 73T^{2} \) |
| 79 | \( 1 + (-6.05 - 10.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.74 + 4.75i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 3.25T + 89T^{2} \) |
| 97 | \( 1 + (4.55 + 7.89i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.00935243958602024511614603005, −9.339354453472194629875615960025, −8.424687026914905621088491607195, −7.48784548549818682095629848357, −6.66660838196899357486294182498, −5.96091345001901944815310780112, −4.90028721063857012513000524765, −4.23128876236959883134571761448, −2.26307694191964373997342136842, −1.52446479141954707100979751003,
0.44980496150069241603862583430, 2.39720366973919982559689031396, 3.70517479695938801032386557591, 4.44484388581840663847271241772, 5.57064046654025726029633112489, 6.39057440179189247757570021477, 6.92893658300723183552476015433, 8.486077552541742581199357587449, 8.886907730399941467997410156539, 10.18003358126041927300478196569