L(s) = 1 | − i·3-s + 0.467i·5-s − 7-s − 9-s + 4.87i·11-s + 4.56i·13-s + 0.467·15-s + 6.09·17-s + 1.34i·19-s + i·21-s + 4.09·23-s + 4.78·25-s + i·27-s − 7.78i·29-s − 4.40·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.208i·5-s − 0.377·7-s − 0.333·9-s + 1.47i·11-s + 1.26i·13-s + 0.120·15-s + 1.47·17-s + 0.308i·19-s + 0.218i·21-s + 0.854·23-s + 0.956·25-s + 0.192i·27-s − 1.44i·29-s − 0.791·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31056 + 0.428807i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31056 + 0.428807i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - 0.467iT - 5T^{2} \) |
| 11 | \( 1 - 4.87iT - 11T^{2} \) |
| 13 | \( 1 - 4.56iT - 13T^{2} \) |
| 17 | \( 1 - 6.09T + 17T^{2} \) |
| 19 | \( 1 - 1.34iT - 19T^{2} \) |
| 23 | \( 1 - 4.09T + 23T^{2} \) |
| 29 | \( 1 + 7.78iT - 29T^{2} \) |
| 31 | \( 1 + 4.40T + 31T^{2} \) |
| 37 | \( 1 - 4.40iT - 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 - 4.15iT - 43T^{2} \) |
| 47 | \( 1 - 6.68T + 47T^{2} \) |
| 53 | \( 1 - 1.34iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 5.49iT - 61T^{2} \) |
| 67 | \( 1 - 5.90iT - 67T^{2} \) |
| 71 | \( 1 - 4.72T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 13.7iT - 83T^{2} \) |
| 89 | \( 1 - 7.96T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−10.47834151472936418585612906344, −9.723643523347426924328954326803, −8.960316342805901739943699759167, −7.74162602681861297069128320253, −7.09413294264565331699204308731, −6.34235846696201103877821376626, −5.15895956576428974340053342752, −4.05164891612955381973817877322, −2.73088562185847073874759392351, −1.49236221692041960328235589386,
0.813460356922776492031875778548, 3.06162852962262700032813783632, 3.53302156970907584264258685596, 5.22075933387531208908932175894, 5.58034990301373412072784717664, 6.85672637921539069142190871546, 7.987331190867382619397088736986, 8.758637795991941172371417518774, 9.504723442987829945549337758234, 10.74644787661730475328474881588