| L(s) = 1 | − 2·2-s + 4·4-s + 5·5-s − 7·7-s − 8·8-s − 10·10-s + 36·11-s − 13·13-s + 14·14-s + 16·16-s − 26·17-s − 47·19-s + 20·20-s − 72·22-s + 99·23-s − 100·25-s + 26·26-s − 28·28-s + 61·29-s − 23·31-s − 32·32-s + 52·34-s − 35·35-s − 50·37-s + 94·38-s − 40·40-s − 70·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.986·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.370·17-s − 0.567·19-s + 0.223·20-s − 0.697·22-s + 0.897·23-s − 4/5·25-s + 0.196·26-s − 0.188·28-s + 0.390·29-s − 0.133·31-s − 0.176·32-s + 0.262·34-s − 0.169·35-s − 0.222·37-s + 0.401·38-s − 0.158·40-s − 0.266·41-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 | \( 1 - p T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 17 | \( 1 + 26 T + p^{3} T^{2} \) |
| 19 | \( 1 + 47 T + p^{3} T^{2} \) |
| 23 | \( 1 - 99 T + p^{3} T^{2} \) |
| 29 | \( 1 - 61 T + p^{3} T^{2} \) |
| 31 | \( 1 + 23 T + p^{3} T^{2} \) |
| 37 | \( 1 + 50 T + p^{3} T^{2} \) |
| 41 | \( 1 + 70 T + p^{3} T^{2} \) |
| 43 | \( 1 + 19 T + p^{3} T^{2} \) |
| 47 | \( 1 + 191 T + p^{3} T^{2} \) |
| 53 | \( 1 + 195 T + p^{3} T^{2} \) |
| 59 | \( 1 + 264 T + p^{3} T^{2} \) |
| 61 | \( 1 - 310 T + p^{3} T^{2} \) |
| 67 | \( 1 + 190 T + p^{3} T^{2} \) |
| 71 | \( 1 - 166 T + p^{3} T^{2} \) |
| 73 | \( 1 - 873 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1191 T + p^{3} T^{2} \) |
| 83 | \( 1 + 259 T + p^{3} T^{2} \) |
| 89 | \( 1 - 635 T + p^{3} T^{2} \) |
| 97 | \( 1 - 133 T + p^{3} T^{2} \) |
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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
−8.790975431458346970772892654277, −7.940153597620560746647625174686, −6.89553350563373769844926435836, −6.46720464085083752048295396892, −5.52115542174638982586087778689, −4.39239498429174279766187696539, −3.33267720737297859268086737274, −2.24864056482737412031567754358, −1.27587852077382299954761649098, 0,
1.27587852077382299954761649098, 2.24864056482737412031567754358, 3.33267720737297859268086737274, 4.39239498429174279766187696539, 5.52115542174638982586087778689, 6.46720464085083752048295396892, 6.89553350563373769844926435836, 7.940153597620560746647625174686, 8.790975431458346970772892654277
