L(s) = 1 | + 2-s + 4-s + (2.11 + 0.716i)5-s + (2.41 − 1.07i)7-s + 8-s + (2.11 + 0.716i)10-s + (−5.13 + 2.96i)11-s + (2.57 + 4.46i)13-s + (2.41 − 1.07i)14-s + 16-s + (1.62 + 0.935i)17-s + (−1.51 + 0.872i)19-s + (2.11 + 0.716i)20-s + (−5.13 + 2.96i)22-s + (−3.20 + 5.55i)23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.947 + 0.320i)5-s + (0.912 − 0.408i)7-s + 0.353·8-s + (0.669 + 0.226i)10-s + (−1.54 + 0.894i)11-s + (0.715 + 1.23i)13-s + (0.645 − 0.288i)14-s + 0.250·16-s + (0.392 + 0.226i)17-s + (−0.346 + 0.200i)19-s + (0.473 + 0.160i)20-s + (−1.09 + 0.632i)22-s + (−0.669 + 1.15i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.402523170\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.402523170\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.11 - 0.716i)T \) |
| 7 | \( 1 + (-2.41 + 1.07i)T \) |
good | 11 | \( 1 + (5.13 - 2.96i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.57 - 4.46i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.62 - 0.935i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.51 - 0.872i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.20 - 5.55i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.16 - 1.25i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4.36iT - 31T^{2} \) |
| 37 | \( 1 + (-1.17 + 0.680i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.79 - 6.57i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.07 + 2.35i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 9.74iT - 47T^{2} \) |
| 53 | \( 1 + (-0.628 + 1.08i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 1.30T + 59T^{2} \) |
| 61 | \( 1 + 11.4iT - 61T^{2} \) |
| 67 | \( 1 + 15.0iT - 67T^{2} \) |
| 71 | \( 1 - 2.88iT - 71T^{2} \) |
| 73 | \( 1 + (1.79 - 3.11i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 8.42T + 79T^{2} \) |
| 83 | \( 1 + (-2.43 - 1.40i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (8.92 + 15.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.23 - 2.13i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.552121272429040606617169260123, −8.339112852755193511582679573605, −7.64656098401232907326571395237, −6.86379198099521846993876722276, −5.99637323802983285248136845848, −5.22167981763215459360837850927, −4.53940709672106937226351918936, −3.52981455445912671234267783123, −2.21091450207953033577389031476, −1.68794330824495931252801112929,
1.00975271476825515309990457119, 2.40886118438940370445564897280, 2.96027561272423992239940180141, 4.38698010500405167012032967959, 5.31765787253711803385138291472, 5.62620612882465913403606648618, 6.39749353933978376397579582184, 7.73931432296678284727744798594, 8.273137231703645040683813939798, 8.910526858701154601864856664334