L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + 2i·7-s + i·8-s − 9-s + i·12-s − i·13-s + 2·14-s + 16-s + i·18-s − 2·19-s + 2·21-s + 6i·23-s + 24-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.755i·7-s + 0.353i·8-s − 0.333·9-s + 0.288i·12-s − 0.277i·13-s + 0.534·14-s + 0.250·16-s + 0.235i·18-s − 0.458·19-s + 0.436·21-s + 1.25i·23-s + 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.467253124\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.467253124\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 4iT - 97T^{2} \) |
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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
−9.205808998921684729461433497612, −8.356890420528798375254142493669, −7.78794684095209089026146896216, −6.71481210974054343056632605705, −5.86710890757803466086088616402, −5.12978148384682678951422762763, −4.04448026027584076842697291472, −2.95977005812852660513515134881, −2.20445428401121700814399015325, −1.02862473838514625318759131743,
0.66196166058861012909453209750, 2.43464846964074873527327815903, 3.73729179735130271383726062099, 4.38496901149212126537299438263, 5.13760116064538577075832622153, 6.20432182140903358784281611310, 6.79072236602258536704916053852, 7.69949874963233030234762750069, 8.461465551553228080475468320071, 9.107828819269495590225103476003