L(s) = 1 | + i·2-s − 4-s + 2i·7-s − i·8-s − 6·11-s + 4i·13-s − 2·14-s + 16-s + 6i·17-s + 4·19-s − 6i·22-s − 4·26-s − 2i·28-s − 6·29-s − 4·31-s + i·32-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.755i·7-s − 0.353i·8-s − 1.80·11-s + 1.10i·13-s − 0.534·14-s + 0.250·16-s + 1.45i·17-s + 0.917·19-s − 1.27i·22-s − 0.784·26-s − 0.377i·28-s − 1.11·29-s − 0.718·31-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.206373 + 0.874212i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.206373 + 0.874212i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 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
−11.46691306363255200295706673721, −10.46532617918269603033219123694, −9.550681759427057251586894441137, −8.588281466561764313932736470777, −7.87067540107579354916680303729, −6.86972012736796772919812145513, −5.72241631195504576264873927835, −5.09655250512672301948311836631, −3.68840496646488921396534688276, −2.15003086525239329527872874140,
0.54501603361966242488037523888, 2.51697367074953278995786915732, 3.48592334877177566588123226046, 4.93617897661492578500048103431, 5.59588701731174473588947707537, 7.45493573375307987980438699169, 7.73966030085716759971920989634, 9.146893401147306246525516267866, 10.02594060123778889785594044970, 10.71188889806956546895894555819