L(s) = 1 | − i·2-s − 3-s − 4-s + i·5-s + i·6-s + 2i·7-s + i·8-s + 9-s + 10-s + 5i·11-s + 12-s + 2·14-s − i·15-s + 16-s + 2·17-s − i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s + 0.755i·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s + 1.50i·11-s + 0.288·12-s + 0.534·14-s − 0.258i·15-s + 0.250·16-s + 0.485·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.431196738\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.431196738\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 5iT - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 11iT - 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 - 11T + 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 15iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 16iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 2iT - 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.442056504694076169330668583022, −7.36379449332286867902451332506, −7.04596303977487781270613693294, −5.87222355214376018319351739419, −5.46792732898408456677176856269, −4.44352471960726017744804309082, −3.95172934677347510245855299813, −2.57310560744625012538482499110, −2.24870790747808629568340072805, −0.909375857703112440824229733450,
0.58381853848495046382908327236, 1.30012115050580688730725716993, 3.05037079576461632985435591999, 3.79976857185222834521731438940, 4.68482692401855903216629956485, 5.32222379829402757503548059111, 6.05743375122740498678500479902, 6.60284507937044046917999370680, 7.43305691806305787979894907511, 8.120528454565688642950939325201