L(s) = 1 | − 7-s − 3·9-s + 2·13-s + 2·17-s + 4·19-s + 23-s + 2·29-s + 2·37-s + 2·41-s − 4·43-s + 12·47-s + 49-s + 2·53-s − 12·59-s + 6·61-s + 3·63-s + 4·67-s + 10·73-s + 4·79-s + 9·81-s + 12·83-s + 6·89-s − 2·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 9-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.208·23-s + 0.371·29-s + 0.328·37-s + 0.312·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.274·53-s − 1.56·59-s + 0.768·61-s + 0.377·63-s + 0.488·67-s + 1.17·73-s + 0.450·79-s + 81-s + 1.31·83-s + 0.635·89-s − 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.17962228065526, −12.43188506749466, −12.11521972771143, −11.79464618521540, −11.11825130750239, −10.82375736319892, −10.38609737739009, −9.655066773931984, −9.405706894732383, −8.891268660027269, −8.405481764353403, −7.894373913042009, −7.528184908951729, −6.848000214957108, −6.385756007546986, −5.914922675804694, −5.418615282732219, −5.047224996184980, −4.333171683499243, −3.572264592548029, −3.423392145635526, −2.624591762942028, −2.316061143978749, −1.273544330124549, −0.8492412130453077, 0,
0.8492412130453077, 1.273544330124549, 2.316061143978749, 2.624591762942028, 3.423392145635526, 3.572264592548029, 4.333171683499243, 5.047224996184980, 5.418615282732219, 5.914922675804694, 6.385756007546986, 6.848000214957108, 7.528184908951729, 7.894373913042009, 8.405481764353403, 8.891268660027269, 9.405706894732383, 9.655066773931984, 10.38609737739009, 10.82375736319892, 11.11825130750239, 11.79464618521540, 12.11521972771143, 12.43188506749466, 13.17962228065526