| L(s) = 1 | − 2·2-s + 4·4-s − 9·5-s − 7·7-s − 8·8-s + 18·10-s − 62·11-s − 13·13-s + 14·14-s + 16·16-s + 16·17-s + 79·19-s − 36·20-s + 124·22-s + 155·23-s − 44·25-s + 26·26-s − 28·28-s − 51·29-s + 243·31-s − 32·32-s − 32·34-s + 63·35-s + 412·37-s − 158·38-s + 72·40-s + 406·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.804·5-s − 0.377·7-s − 0.353·8-s + 0.569·10-s − 1.69·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.228·17-s + 0.953·19-s − 0.402·20-s + 1.20·22-s + 1.40·23-s − 0.351·25-s + 0.196·26-s − 0.188·28-s − 0.326·29-s + 1.40·31-s − 0.176·32-s − 0.161·34-s + 0.304·35-s + 1.83·37-s − 0.674·38-s + 0.284·40-s + 1.54·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 + 9 T + p^{3} T^{2} \) |
| 11 | \( 1 + 62 T + p^{3} T^{2} \) |
| 17 | \( 1 - 16 T + p^{3} T^{2} \) |
| 19 | \( 1 - 79 T + p^{3} T^{2} \) |
| 23 | \( 1 - 155 T + p^{3} T^{2} \) |
| 29 | \( 1 + 51 T + p^{3} T^{2} \) |
| 31 | \( 1 - 243 T + p^{3} T^{2} \) |
| 37 | \( 1 - 412 T + p^{3} T^{2} \) |
| 41 | \( 1 - 406 T + p^{3} T^{2} \) |
| 43 | \( 1 + 103 T + p^{3} T^{2} \) |
| 47 | \( 1 + 429 T + p^{3} T^{2} \) |
| 53 | \( 1 - 169 T + p^{3} T^{2} \) |
| 59 | \( 1 + 320 T + p^{3} T^{2} \) |
| 61 | \( 1 + 614 T + p^{3} T^{2} \) |
| 67 | \( 1 - 258 T + p^{3} T^{2} \) |
| 71 | \( 1 - 264 T + p^{3} T^{2} \) |
| 73 | \( 1 + 121 T + p^{3} T^{2} \) |
| 79 | \( 1 + 967 T + p^{3} T^{2} \) |
| 83 | \( 1 - 679 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1059 T + p^{3} T^{2} \) |
| 97 | \( 1 + 21 T + p^{3} T^{2} \) |
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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
−8.499970430518192937434054774627, −7.67730852035444639423588825345, −7.49427064250290607775544949669, −6.31879624562593299830106840822, −5.36700794902767348320746914302, −4.47936544247226284264361702468, −3.15334832340702267483542245949, −2.60145922115317657998030684961, −0.982560490204876454984949843947, 0,
0.982560490204876454984949843947, 2.60145922115317657998030684961, 3.15334832340702267483542245949, 4.47936544247226284264361702468, 5.36700794902767348320746914302, 6.31879624562593299830106840822, 7.49427064250290607775544949669, 7.67730852035444639423588825345, 8.499970430518192937434054774627
