L(s) = 1 | − 2.44i·3-s + (0.224 + 2.22i)5-s + i·7-s − 2.99·9-s − 4.89·11-s + 0.449i·13-s + (5.44 − 0.550i)15-s − 2i·17-s + 6.44·19-s + 2.44·21-s − 6.89i·23-s + (−4.89 + i)25-s − 2.89·29-s − 0.898·31-s + 11.9i·33-s + ⋯ |
L(s) = 1 | − 1.41i·3-s + (0.100 + 0.994i)5-s + 0.377i·7-s − 0.999·9-s − 1.47·11-s + 0.124i·13-s + (1.40 − 0.142i)15-s − 0.485i·17-s + 1.47·19-s + 0.534·21-s − 1.43i·23-s + (−0.979 + 0.200i)25-s − 0.538·29-s − 0.161·31-s + 2.08i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6423331709\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6423331709\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.224 - 2.22i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + 2.44iT - 3T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 0.449iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 6.44T + 19T^{2} \) |
| 23 | \( 1 + 6.89iT - 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 + 0.898T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 8.89iT - 43T^{2} \) |
| 47 | \( 1 - 0.898iT - 47T^{2} \) |
| 53 | \( 1 - 1.10iT - 53T^{2} \) |
| 59 | \( 1 + 6.44T + 59T^{2} \) |
| 61 | \( 1 + 8.44T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 6.89iT - 73T^{2} \) |
| 79 | \( 1 - 2.89T + 79T^{2} \) |
| 83 | \( 1 + 2.44iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 3.79iT - 97T^{2} \) |
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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
−8.373934939769084617082117143190, −7.65839401197566988377979779497, −7.22040433431759214794449229680, −6.47496136319428939972397312722, −5.70698439708876773513935393030, −4.88704124637001824397221639204, −3.25967745697196948144728411301, −2.61930405467392647325283009475, −1.78669590156165920953865237500, −0.21258566062777738064442856662,
1.47458733684502290564505463431, 3.04585413755205914392059975522, 3.77600919673265572612566353278, 4.74406624879041187935813394233, 5.25959068825765931601914490896, 5.81708238131126741238473586737, 7.37599353124371915241142240342, 7.943263763056273480102716517043, 8.805957205176426531800976940500, 9.611143014312194736476344211335