L(s) = 1 | + (0.258 − 0.965i)3-s + (−0.707 + 0.707i)5-s + (0.866 − 0.5i)7-s + (−0.866 − 0.499i)9-s + (−0.965 − 1.67i)11-s + (0.500 + 0.866i)15-s + (−0.258 − 0.965i)21-s − 1.00i·25-s + (−0.707 + 0.707i)27-s − 0.517i·29-s + (−0.866 + 0.5i)31-s + (−1.86 + 0.500i)33-s + (−0.258 + 0.965i)35-s + (0.965 − 0.258i)45-s + (0.499 − 0.866i)49-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)3-s + (−0.707 + 0.707i)5-s + (0.866 − 0.5i)7-s + (−0.866 − 0.499i)9-s + (−0.965 − 1.67i)11-s + (0.500 + 0.866i)15-s + (−0.258 − 0.965i)21-s − 1.00i·25-s + (−0.707 + 0.707i)27-s − 0.517i·29-s + (−0.866 + 0.5i)31-s + (−1.86 + 0.500i)33-s + (−0.258 + 0.965i)35-s + (0.965 − 0.258i)45-s + (0.499 − 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8982038976\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8982038976\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
good | 11 | \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + 0.517iT - T^{2} \) |
| 31 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.366 + 0.366i)T - iT^{2} \) |
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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
−8.379135740400703814642522903725, −7.65655044144408990972026174915, −7.32852204469118780557139505896, −6.31055998223499601769795588757, −5.68864435433801558630133753895, −4.65332136198908728273540297918, −3.48501601351933442731974813860, −2.98485407401563782897383173469, −1.85552736412119374360199749842, −0.49696285530254795849518448935,
1.76124191434841000765943120228, 2.68330665081589269561858035395, 3.84515637862489627195311318068, 4.61789140480253257575977642772, 5.01956705850234686379004978122, 5.69908656418861469216120801088, 7.15737707772892731653174114857, 7.80712169102451728596012081660, 8.336462470415157336173521998503, 9.098294708655223576783801678892