| L(s) = 1 | + 3·2-s + 4-s + 12·5-s − 12·7-s − 21·8-s + 36·10-s + 66·13-s − 36·14-s − 71·16-s − 114·17-s − 42·19-s + 12·20-s − 18·23-s + 19·25-s + 198·26-s − 12·28-s + 186·29-s − 308·31-s − 45·32-s − 342·34-s − 144·35-s − 146·37-s − 126·38-s − 252·40-s + 42·41-s + 366·43-s − 54·46-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1/8·4-s + 1.07·5-s − 0.647·7-s − 0.928·8-s + 1.13·10-s + 1.40·13-s − 0.687·14-s − 1.10·16-s − 1.62·17-s − 0.507·19-s + 0.134·20-s − 0.163·23-s + 0.151·25-s + 1.49·26-s − 0.0809·28-s + 1.19·29-s − 1.78·31-s − 0.248·32-s − 1.72·34-s − 0.695·35-s − 0.648·37-s − 0.537·38-s − 0.996·40-s + 0.159·41-s + 1.29·43-s − 0.173·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 66 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 + 42 T + p^{3} T^{2} \) |
| 23 | \( 1 + 18 T + p^{3} T^{2} \) |
| 29 | \( 1 - 186 T + p^{3} T^{2} \) |
| 31 | \( 1 + 308 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 42 T + p^{3} T^{2} \) |
| 43 | \( 1 - 366 T + p^{3} T^{2} \) |
| 47 | \( 1 + 618 T + p^{3} T^{2} \) |
| 53 | \( 1 - 408 T + p^{3} T^{2} \) |
| 59 | \( 1 - 132 T + p^{3} T^{2} \) |
| 61 | \( 1 + 630 T + p^{3} T^{2} \) |
| 67 | \( 1 + 452 T + p^{3} T^{2} \) |
| 71 | \( 1 - 282 T + p^{3} T^{2} \) |
| 73 | \( 1 + 684 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1272 T + p^{3} T^{2} \) |
| 83 | \( 1 + 432 T + p^{3} T^{2} \) |
| 89 | \( 1 + 954 T + p^{3} T^{2} \) |
| 97 | \( 1 - 326 T + p^{3} T^{2} \) |
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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
−8.994843914743951221101935483288, −8.596356318810173680753919017784, −6.94427527513079405464318448646, −6.18316284092337417489211352823, −5.80173663768574755323855184176, −4.66854448928458628169696553925, −3.83541396049379968015384410688, −2.84850810699099482663415166216, −1.74655279970249729963429530969, 0,
1.74655279970249729963429530969, 2.84850810699099482663415166216, 3.83541396049379968015384410688, 4.66854448928458628169696553925, 5.80173663768574755323855184176, 6.18316284092337417489211352823, 6.94427527513079405464318448646, 8.596356318810173680753919017784, 8.994843914743951221101935483288
