L(s) = 1 | − 2·5-s + 2·13-s − 2·17-s + 19-s − 25-s − 2·29-s + 4·31-s + 2·37-s − 6·41-s + 4·43-s − 7·49-s − 10·53-s − 4·59-s − 2·61-s − 4·65-s + 12·67-s − 6·73-s + 4·79-s − 8·83-s + 4·85-s − 6·89-s − 2·95-s − 14·97-s + 6·101-s − 4·103-s − 20·107-s − 6·109-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.554·13-s − 0.485·17-s + 0.229·19-s − 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.328·37-s − 0.937·41-s + 0.609·43-s − 49-s − 1.37·53-s − 0.520·59-s − 0.256·61-s − 0.496·65-s + 1.46·67-s − 0.702·73-s + 0.450·79-s − 0.878·83-s + 0.433·85-s − 0.635·89-s − 0.205·95-s − 1.42·97-s + 0.597·101-s − 0.394·103-s − 1.93·107-s − 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.252030560408320635093882963616, −7.87537518402846875743837031097, −6.92822227886234169606129538581, −6.25452895745643782338001477919, −5.27360891135767646869901082305, −4.37266688783284321332139300895, −3.68712589588092002834188844808, −2.75741804606621082589758524445, −1.44248876304714278885152632746, 0,
1.44248876304714278885152632746, 2.75741804606621082589758524445, 3.68712589588092002834188844808, 4.37266688783284321332139300895, 5.27360891135767646869901082305, 6.25452895745643782338001477919, 6.92822227886234169606129538581, 7.87537518402846875743837031097, 8.252030560408320635093882963616