L(s) = 1 | − 2·2-s + 4·4-s − 3·5-s + 7·7-s − 8·8-s + 6·10-s + 54·11-s + 13·13-s − 14·14-s + 16·16-s + 96·17-s − 151·19-s − 12·20-s − 108·22-s − 33·23-s − 116·25-s − 26·26-s + 28·28-s − 183·29-s − 331·31-s − 32·32-s − 192·34-s − 21·35-s − 88·37-s + 302·38-s + 24·40-s + 42·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.268·5-s + 0.377·7-s − 0.353·8-s + 0.189·10-s + 1.48·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.36·17-s − 1.82·19-s − 0.134·20-s − 1.04·22-s − 0.299·23-s − 0.927·25-s − 0.196·26-s + 0.188·28-s − 1.17·29-s − 1.91·31-s − 0.176·32-s − 0.968·34-s − 0.101·35-s − 0.391·37-s + 1.28·38-s + 0.0948·40-s + 0.159·41-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 | \( 1 + 3 T + p^{3} T^{2} \) |
| 11 | \( 1 - 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 96 T + p^{3} T^{2} \) |
| 19 | \( 1 + 151 T + p^{3} T^{2} \) |
| 23 | \( 1 + 33 T + p^{3} T^{2} \) |
| 29 | \( 1 + 183 T + p^{3} T^{2} \) |
| 31 | \( 1 + 331 T + p^{3} T^{2} \) |
| 37 | \( 1 + 88 T + p^{3} T^{2} \) |
| 41 | \( 1 - 42 T + p^{3} T^{2} \) |
| 43 | \( 1 - 353 T + p^{3} T^{2} \) |
| 47 | \( 1 - 465 T + p^{3} T^{2} \) |
| 53 | \( 1 + 195 T + p^{3} T^{2} \) |
| 59 | \( 1 + 552 T + p^{3} T^{2} \) |
| 61 | \( 1 - 470 T + p^{3} T^{2} \) |
| 67 | \( 1 - 254 T + p^{3} T^{2} \) |
| 71 | \( 1 + 132 T + p^{3} T^{2} \) |
| 73 | \( 1 + 943 T + p^{3} T^{2} \) |
| 79 | \( 1 + 727 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1197 T + p^{3} T^{2} \) |
| 89 | \( 1 + 753 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1037 T + p^{3} T^{2} \) |
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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
−8.807778038715973021333280679946, −7.79719022622809512024979568894, −7.26926588472458538073084200229, −6.22045101214396853082940826840, −5.61466368032343484444658193040, −4.14001728350096255951137313476, −3.63998825054373724883435038736, −2.10844303475684512184237462396, −1.30600100397476910703189075135, 0,
1.30600100397476910703189075135, 2.10844303475684512184237462396, 3.63998825054373724883435038736, 4.14001728350096255951137313476, 5.61466368032343484444658193040, 6.22045101214396853082940826840, 7.26926588472458538073084200229, 7.79719022622809512024979568894, 8.807778038715973021333280679946