L(s) = 1 | − i·2-s + 2i·3-s − 4-s + 2·6-s + i·7-s + i·8-s − 9-s − 2i·12-s + 4i·13-s + 14-s + 16-s + 6i·17-s + i·18-s − 2·19-s − 2·21-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.15i·3-s − 0.5·4-s + 0.816·6-s + 0.377i·7-s + 0.353i·8-s − 0.333·9-s − 0.577i·12-s + 1.10i·13-s + 0.267·14-s + 0.250·16-s + 1.45i·17-s + 0.235i·18-s − 0.458·19-s − 0.436·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00849 + 0.623283i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00849 + 0.623283i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 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
−11.48583316339616217251470155317, −10.60105785382614769177464695896, −10.00163421698977404390334482327, −9.049577387658908266269627866472, −8.393707496562347182363489127918, −6.75392696237221147300912939967, −5.45836858458498125749419368293, −4.35574163478167456399141262308, −3.63535422454790820219193927071, −2.01314549923338325550520851977,
0.875332837139987971235460555126, 2.78433191336991662037945800384, 4.47992619641238253053421396260, 5.70856463881718166521226143434, 6.70228708980095922945847101253, 7.48736582630619125405222246396, 8.104838269498584631069763936761, 9.296527828953851678438374022882, 10.33953066651889968860924777574, 11.46975453226636471827058596761