L(s) = 1 | + (0.345 + 0.345i)2-s + (−0.805 − 0.805i)3-s − 1.76i·4-s + (−2.18 + 0.490i)5-s − 0.556i·6-s + (2.06 + 2.06i)7-s + (1.29 − 1.29i)8-s − 1.70i·9-s + (−0.922 − 0.583i)10-s + (−1.41 + 1.41i)12-s + (2.06 − 2.06i)13-s + 1.42i·14-s + (2.15 + 1.36i)15-s − 2.62·16-s + (−3.72 − 3.72i)17-s + (0.587 − 0.587i)18-s + ⋯ |
L(s) = 1 | + (0.244 + 0.244i)2-s + (−0.465 − 0.465i)3-s − 0.880i·4-s + (−0.975 + 0.219i)5-s − 0.227i·6-s + (0.779 + 0.779i)7-s + (0.459 − 0.459i)8-s − 0.567i·9-s + (−0.291 − 0.184i)10-s + (−0.409 + 0.409i)12-s + (0.573 − 0.573i)13-s + 0.380i·14-s + (0.555 + 0.351i)15-s − 0.656·16-s + (−0.903 − 0.903i)17-s + (0.138 − 0.138i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.364032 - 0.797404i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.364032 - 0.797404i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.18 - 0.490i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.345 - 0.345i)T + 2iT^{2} \) |
| 3 | \( 1 + (0.805 + 0.805i)T + 3iT^{2} \) |
| 7 | \( 1 + (-2.06 - 2.06i)T + 7iT^{2} \) |
| 13 | \( 1 + (-2.06 + 2.06i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.72 + 3.72i)T + 17iT^{2} \) |
| 19 | \( 1 + 4.09T + 19T^{2} \) |
| 23 | \( 1 + (2.12 + 2.12i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.64T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 + (-0.822 + 0.822i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.55iT - 41T^{2} \) |
| 43 | \( 1 + (-5.07 + 5.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.60 - 2.60i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.04 - 1.04i)T + 53iT^{2} \) |
| 59 | \( 1 + 1.60iT - 59T^{2} \) |
| 61 | \( 1 - 6.93iT - 61T^{2} \) |
| 67 | \( 1 + (-1.31 + 1.31i)T - 67iT^{2} \) |
| 71 | \( 1 - 2.87T + 71T^{2} \) |
| 73 | \( 1 + (-10.4 + 10.4i)T - 73iT^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + (-4.72 + 4.72i)T - 83iT^{2} \) |
| 89 | \( 1 - 11.1iT - 89T^{2} \) |
| 97 | \( 1 + (-7.11 + 7.11i)T - 97iT^{2} \) |
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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
−10.79590774757326125355467316973, −9.316211775566107937910359139178, −8.579245430091363723070319941616, −7.47213751918136902259537620822, −6.63196658234919471653227618797, −5.82037488788201848144557793730, −4.91283063382061315906858942745, −3.83705277468178342951083421130, −2.10684142602976516850681637972, −0.47036276895506745524001796732,
1.96113431573570686827673205155, 3.82108409522325554337362881181, 4.19123476431040735639791759086, 5.04031093887339567003333847054, 6.57407986342850975244092189770, 7.70361167023791057737059564687, 8.135044764775652018507409941513, 9.064511470387725263174932623561, 10.59649942366885964335123782065, 11.16247786154390228439797232509