L(s) = 1 | + (0.866 − 0.5i)3-s + (0.866 − 2.5i)7-s + (−1 + 1.73i)9-s + (−3 − 5.19i)11-s + 2i·13-s + (5.19 − 3i)17-s + (4 − 6.92i)19-s + (−0.500 − 2.59i)21-s + (−2.59 − 1.5i)23-s + 5i·27-s − 3·29-s + (−1 − 1.73i)31-s + (−5.19 − 3i)33-s + (6.92 + 4i)37-s + (1 + 1.73i)39-s + ⋯ |
L(s) = 1 | + (0.499 − 0.288i)3-s + (0.327 − 0.944i)7-s + (−0.333 + 0.577i)9-s + (−0.904 − 1.56i)11-s + 0.554i·13-s + (1.26 − 0.727i)17-s + (0.917 − 1.58i)19-s + (−0.109 − 0.566i)21-s + (−0.541 − 0.312i)23-s + 0.962i·27-s − 0.557·29-s + (−0.179 − 0.311i)31-s + (−0.904 − 0.522i)33-s + (1.13 + 0.657i)37-s + (0.160 + 0.277i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26519 - 1.04128i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26519 - 1.04128i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.866 + 2.5i)T \) |
good | 3 | \( 1 + (-0.866 + 0.5i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + (-5.19 + 3i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.92 - 4i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 5iT - 43T^{2} \) |
| 47 | \( 1 + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.3 + 6i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3iT - 83T^{2} \) |
| 89 | \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−10.30099858917475190563373019992, −9.356606248631007618744292892105, −8.313689378981753794526519773166, −7.77391855377099544871682595950, −6.98605285245511176583640423195, −5.65052428073291418042029720139, −4.85594022073743344940773153822, −3.44523461624446216444515653620, −2.57691839692257883958147439744, −0.838381786191964814041901911863,
1.82369861939299468177614442616, 3.00430670907121122626317071299, 4.02787335606104075141367167746, 5.41237621230457878230579942949, 5.86278475541282963043007932529, 7.49935966474912430085120076445, 7.979823113873477978417731740519, 8.942172801613675109721939521145, 9.941646545808340282293226632932, 10.20131039444586627625302703365