L(s) = 1 | + i·2-s + (0.707 + 0.707i)3-s − 4-s + (1.84 − 1.26i)5-s + (−0.707 + 0.707i)6-s + (1.64 + 1.64i)7-s − i·8-s + 1.00i·9-s + (1.26 + 1.84i)10-s − 4.17i·11-s + (−0.707 − 0.707i)12-s + 1.20i·13-s + (−1.64 + 1.64i)14-s + (2.19 + 0.410i)15-s + 16-s − 0.221·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (0.824 − 0.565i)5-s + (−0.288 + 0.288i)6-s + (0.620 + 0.620i)7-s − 0.353i·8-s + 0.333i·9-s + (0.399 + 0.583i)10-s − 1.25i·11-s + (−0.204 − 0.204i)12-s + 0.332i·13-s + (−0.438 + 0.438i)14-s + (0.567 + 0.105i)15-s + 0.250·16-s − 0.0538·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.246259828\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.246259828\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1.84 + 1.26i)T \) |
| 37 | \( 1 + (0.388 - 6.07i)T \) |
good | 7 | \( 1 + (-1.64 - 1.64i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.17iT - 11T^{2} \) |
| 13 | \( 1 - 1.20iT - 13T^{2} \) |
| 17 | \( 1 + 0.221T + 17T^{2} \) |
| 19 | \( 1 + (-3.73 - 3.73i)T + 19iT^{2} \) |
| 23 | \( 1 - 4.24iT - 23T^{2} \) |
| 29 | \( 1 + (-5.45 + 5.45i)T - 29iT^{2} \) |
| 31 | \( 1 + (-1.77 - 1.77i)T + 31iT^{2} \) |
| 41 | \( 1 + 2.58iT - 41T^{2} \) |
| 43 | \( 1 + 7.69iT - 43T^{2} \) |
| 47 | \( 1 + (-5.62 - 5.62i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.45 - 4.45i)T - 53iT^{2} \) |
| 59 | \( 1 + (3.90 + 3.90i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.901 - 0.901i)T + 61iT^{2} \) |
| 67 | \( 1 + (10.3 - 10.3i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.11T + 71T^{2} \) |
| 73 | \( 1 + (-3.76 - 3.76i)T + 73iT^{2} \) |
| 79 | \( 1 + (0.571 + 0.571i)T + 79iT^{2} \) |
| 83 | \( 1 + (-7.69 + 7.69i)T - 83iT^{2} \) |
| 89 | \( 1 + (4.28 - 4.28i)T - 89iT^{2} \) |
| 97 | \( 1 + 8.59T + 97T^{2} \) |
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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
−9.730227806183906968491683982475, −9.030386603325592895640175433333, −8.408083525518384953984566354754, −7.79928212179701663208793327590, −6.44309184007089106610679094659, −5.64181562843933767696889624827, −5.10978261274917169419347755316, −3.99439556505434249659486823713, −2.76173222857524188282477385001, −1.34059913720409397649616791176,
1.19656085811668219926701581462, 2.25194085343269019983200956478, 3.08193655727293908360731913421, 4.41009056933162341109873348531, 5.18308399387487393302693442317, 6.50599725126450241137013704922, 7.23812888189914437083977956201, 8.018800987973081455411039215900, 9.125977141608364220394286065571, 9.734632208966411432985511337217