L(s) = 1 | − 3·9-s − 4·13-s − 8·17-s + 10·29-s − 12·37-s − 10·41-s − 7·49-s + 4·53-s + 10·61-s − 16·73-s + 9·81-s − 10·89-s + 8·97-s − 2·101-s + 6·109-s + 16·113-s + 12·117-s + ⋯ |
L(s) = 1 | − 9-s − 1.10·13-s − 1.94·17-s + 1.85·29-s − 1.97·37-s − 1.56·41-s − 49-s + 0.549·53-s + 1.28·61-s − 1.87·73-s + 81-s − 1.05·89-s + 0.812·97-s − 0.199·101-s + 0.574·109-s + 1.50·113-s + 1.10·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.911668703232740574424658064818, −8.772598804940102831000750865243, −8.422643539109821102866019859152, −7.09375449823698285987770578672, −6.47631929163503178303876770385, −5.26604485491629488374008831599, −4.52063183990333424217701740419, −3.11970456220302029413015085689, −2.12852635758913165860364844653, 0,
2.12852635758913165860364844653, 3.11970456220302029413015085689, 4.52063183990333424217701740419, 5.26604485491629488374008831599, 6.47631929163503178303876770385, 7.09375449823698285987770578672, 8.422643539109821102866019859152, 8.772598804940102831000750865243, 9.911668703232740574424658064818