L(s) = 1 | + (1.60 − 2.77i)3-s + (−1.53 − 2.15i)7-s + (−3.63 − 6.29i)9-s + (−2.10 + 3.64i)11-s + 0.204·13-s + (2.53 − 4.38i)17-s + (−0.531 − 0.921i)19-s + (−8.44 + 0.794i)21-s + (−1.07 − 1.85i)23-s − 13.6·27-s − 7.47·29-s + (4.23 − 7.33i)31-s + (6.73 + 11.6i)33-s + (5.30 + 9.19i)37-s + (0.327 − 0.567i)39-s + ⋯ |
L(s) = 1 | + (0.925 − 1.60i)3-s + (−0.579 − 0.815i)7-s + (−1.21 − 2.09i)9-s + (−0.633 + 1.09i)11-s + 0.0566·13-s + (0.614 − 1.06i)17-s + (−0.122 − 0.211i)19-s + (−1.84 + 0.173i)21-s + (−0.223 − 0.386i)23-s − 2.63·27-s − 1.38·29-s + (0.760 − 1.31i)31-s + (1.17 + 2.03i)33-s + (0.872 + 1.51i)37-s + (0.0524 − 0.0908i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.444473 - 1.56296i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.444473 - 1.56296i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.53 + 2.15i)T \) |
good | 3 | \( 1 + (-1.60 + 2.77i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (2.10 - 3.64i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.204T + 13T^{2} \) |
| 17 | \( 1 + (-2.53 + 4.38i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.531 + 0.921i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.07 + 1.85i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 + (-4.23 + 7.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.30 - 9.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 8.26T + 43T^{2} \) |
| 47 | \( 1 + (1.63 + 2.83i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.83 - 4.91i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.602 - 1.04i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.827 + 1.43i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.20 + 10.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.591T + 71T^{2} \) |
| 73 | \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.27 + 5.67i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.88T + 83T^{2} \) |
| 89 | \( 1 + (-4.63 - 8.02i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 1.33T + 97T^{2} \) |
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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
−9.753130603659561728218017618543, −9.312258049400789139754459285128, −7.88185165257663655207745893084, −7.63151391480722183902930694956, −6.84191294309327227576678257116, −5.96050300781028668919832306540, −4.38233950449045887328237822135, −3.06531314115646709148416392370, −2.20723244531936496305308864821, −0.74930157056138887977026048162,
2.45673508098549560074116577125, 3.34377745440858729807274957963, 4.09005112590600469218562763895, 5.44564718522951509743082422360, 5.92194869720933228754254768241, 7.74983013597651129037095299592, 8.460267166583342343322980004742, 9.159090047870481032193923559066, 9.805013279388424390365086020820, 10.68020541451078475256931799236